Josephson Tunneling at the Atomic Scale: The Josephson ... · Josephson Tunneling at the Atomic...

POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES

acceptée sur proposition du jury:

Prof. A. Pautz, président du juryProf. K. Kern, directeur de thèse

Dr F. Portier, rapporteurProf. H. J. Suderow, rapporteur

Prof. D. Pavuna, rapporteur

Josephson Tunneling at the Atomic Scale: The Josephson Effect as a Local Probe

THÈSE NO 6750 (2015)

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE

PRÉSENTÉE LE 25 AOÛT 2015

À LA FACULTÉ DES SCIENCES DE BASELABORATOIRE DE SCIENCE À L'ÉCHELLE NANOMÉTRIQUE

PROGRAMME DOCTORAL EN PHYSIQUE

Suisse2015

PAR

Berthold JÄCK

Non est ad astra mollis e terris via

— Lucius Annaeus Seneca

AcknowledgementsIt took me about three and a half years, with many ups and downs, to complete this thesis. At

this point, I want to say thank you to all the people that supported me on my way by any means.

First of all, I am grateful to Prof. Klaus Kern for giving me the opportunity to perform my

doctoral studies under his supervision at the Max-Planck-Institute for Solid State Research in

Stuttgart, Germany. For the entire time of my studies, I enjoyed a significant scientific freedom,

such that I had the great possibility to pursue and realize my own ideas. In particular, I want

to acknowledge his personal support, which allowed me to attend conferences, workshops

and schools, have interesting scientific discussions as well as a wonderful time in the whole of

Europe and across the USA. I highly appreciate his support and the trust he put in me.

I am very grateful to Dr. Christian Ast and Dr. Markus Etzkorn for supervising my stud-

ies, their continuous support and many fruitful discussions about physics. I highly benefited

and learned from their exceptional experimental, theoretical and computational knowledge

as well as experience.

I am thankful to Prof. Andreas Pautz, Prof. Davor Pavuna, Prof. Hermann Suderow and

Dr. Fabien Portier for being part of my thesis committee.

Special thanks go to my fellow PhD colleague Matthias Eltschka. We really worked hard

to tame the beast, a.k.a. the milli-Kelvin STM, and spent many day and night shifts together in

the MPI. Our shared working ethics certainly helped to accomplish this thesis.

All the results that I presented in this thesis would not exist without the permanent help

and support from many people at the MPI. In particular, I am grateful to our technicians

Wolfgang Stiepany, Peter Andler and Marco Memmler for their permanent support on taming

the beast and their immediate help on repairing parts that I had damaged. Moreover, they

were also great football team mates in Kernspaltung. I am also very thankful to Sabine Birtel

who always had a helping hand for me. At this point, I also want to express my thanks to the

low temperature service group, the IT service group, the crystal growth service group and the

technology service group of our institute.

I am very thankful to Dr. Christian Urbina and Dr. Fabien Portier from CEA Saclay. I benefited

a lot from their comprehensive knowledge on Josephson physics and I highly appreciated

i

Acknowledgements

the opportunity to visit the Quantronics group at the CEA Saclay. I also want to express my

gratitude to Prof. Joachim Ankerhold, Prof. Alex Balatsky, Prof. Gerd-Ludwig Ingold, Prof.

Dimitri Roditchev and Prof. Alexey Ustinov for fruitful discussions about Josephson physics.

I am very grateful to Dr. Thomas White for thoroughly proofreading my thesis within the

shortest possible time.

Working on a PhD thesis in experimental physics focuses on lab work, data analysis and

scientific discussion. However, this job certainly runs better if you are in good company, which

I certainly was. I’m grateful to have met Dr. Laura Rizzi and Daniel Weber in the guesthouse

on my first day here in Stuttgart. Their company gave me an easy start and I’m very happy to

be still in friendly contact with Dan. Special thanks go to my ol’ 6B10 office crew, Alexander

’Ali’ Kölker and Jon ’Nathan’ Parnell, for fun times, good music and serious target shooting

competitions. I also highly enjoyed my “scientific“ roadtrips through Europe and the United

States together with Daniel Rosenblatt and Christoph Grosse. I am glad that I met Sebastian

Koslowski and that we share more than only our passion for running. Moreover, I want to

thank Carola Straßer, Verena Schendel, Lukas Schlipf, Claudius Morchutt, Christian Dette, Ivan

Pentegov, Matthias Muenks, Jacob Senkpiel, Dr. Tobias Herden, Benjamin Wurster, and Diana

Hoetger for fun times at the MPI, on the Cannstatter Wasen and anywhere else.

I want to express my deepest thankfulness to my parents, Dr. Herbert Jäck and Dorothea

Jäck who, always and literally, unconditionally supported my path of life. My highest gratitude

goes to my partner Silvia who cheered me up in hard times and enjoyed the happy times with

me. Her permanent support was outstanding, such that it is not wrong to say: She did half of

all the work!

Stuttgart, 21st of May, 2015 B. J.

ii

AbstractThe aim of this thesis is to characterize the properties of a Josephson junction in a Scanning

Tunneling Microscope (STM) at millikelvin temperatures and to implement Josephson STM

(JSTM) as a versatile probe at the atomic scale. To this end we investigate the I (VJ) tunnel-

ing characteristics of the Josephson junction in our STM at a base temperature of 15 mK

by means of current-biased and voltage-biased experiments. We find that in the tunneling

regime, GN ¿ G0, the Josephson junction is operated in the dynamical Coulomb blockade

(DCB) regime in which the sequential tunneling of Cooper pairs dominates the tunneling

current. Employing P (E)-theory allows us to model experimental I (VJ) characteristics from

voltage-biased experiments and determine experimental values of the Josephson critical

current in agreement to theory. Moreover, we observe a breakdown of P (E)-theory for ex-

periments at large values of the tunneling conductance GN ≈G0, which could indicate that

the coherent tunneling of Cooper pairs strongly contributes to the tunneling current in this

limit. We also observe that the Josephson junction in an STM at temperatures well below

100 mK is highly sensitive to its electromagnetic environment that results from its tiny junc-

tion capacitance of a few femtofarads. The combination of the experimental results with

numerical simulations reveals that the immediate environment of the Josephson junction

in an STM is frequency-dependent and, additionally, that a typical STM geometry shares

the electromagnetic properties of a monopole antenna with the STM tip itself acting as the

antenna rod. Comparing the I (VJ) curves of voltage-biased and current-biased experiments,

we observe that the time evolution of the junction phase is strongly effected by dissipation

due to quasiparticle excitations. From investigations on the retrapping current IR, we show

first that the temporal evolution of the junction phase in our STM satisfies a classical equation

of motion. Second, we can determine two different channels for energy dissipation of the

junction phase. For tunneling resistance values of RN ≤ 150kΩ the junction dissipates via

Andreev reflections, whereas for larger values of RN ≥ 150kΩ energy dissipation is dominated

by the lifetime effects of Cooper pairs. Moreover, from comparing both experiments we also

observe that the quantum-mechanical nature of the junction phase manifests in quantum-

mechanical phenomena, such as phase tunneling, which strongly alter our experimental I (VJ)

characteristics for GN ≤G0.

To conclude, within this thesis we characterized the properties of a Josephson junction in an

STM that is operated at millikelvin temperatures. Hence, this work represents necessary and

fundamental steps that allows us to employ the Josephson effect as a versatile probe on the

atomic scale.

iii

Acknowledgements

Key words: Josephson Effect, Scanning Tunneling Microscopy (STM), Josephson Scanning

Tunneling Microscopy (JSTM), Dynamical Coulomb Blockade, Quasiparticle Dissipation

iv

ZusammenfassungDas Ziel der vorliegenden Dissertation ist zunächst die Eigenschaften eines Josephson Tunnel-

kontakts in einem Rastertunnelmikroskop (RTM) bei Millikelvin-Temperaturen zu bestimmen

und dadurch Josephson RTM (JRTM) als eine vielseitig einsetzbare Sonde auf atomarer Ebene

zu etablieren. Hierfür untersuchten wir die Eigenschaften der I (VJ)-Spektren des Josephson

Tunnelkontaktes in unserem RTM bei einer Basistemperatur von 15 mK mittels strom- und

spanunnungsgetriebener Experimente. Für den Tunnelbereich, GN ¿G0, ergaben die Unter-

suchungen, dass sich der Josephson Tunnelkontakt innerhalb des dynamischen Coulomb-

Blockade (DCB) Regimes befindet, in welchem das sequentielle Tunneln von Cooper-Paaren

den Tunnelstrom dominiert. Mit Hilfe der sogenannten P (E )-Theorie konnten wir sowohl die

experimentellen I (VJ)-Spektren aus spannungsgetriebenen Experimenten fitten als auch ex-

perimentelle Werte des kritischen Josephson Stromes bestimmen. Darüberhinaus stellten wir

fest, dass diese experimentellen Werte mit thereotisch berechneten Werten übereinstimmen.

Des Weiteren ergaben unsere Untersuchungen, dass die bei großen Leitfähigkeitswerten des

Tunnelkontakts, GN ≈G0, gemessenen Cooper-Paar Tunnelspektren nicht mehr mit Hilfe der

P (E )-Theorie beschrieben werden können. Dies könnte darauf hinweisen, dass das kohärente

Tunneln der Copper-Paare wesentlich zu dem Tunnelstrom unter dieser Bedingung beiträgt.

Wir beobachteten auch, dass der Josephson Effekt in einem STM bei Temperaturen von

deutlich weniger als 100 mK höchst empfindsam gegenüber seiner elektromagnetischen Um-

gebung ist, was von der äußerst geringen Tunnelbarrierenkapaztität von wenigen Femtofarad

herrührt. Der Vergleich der experimentellen Daten mit einer numerischen Simulation zeigte,

dass die unmittelbare Umgebung des Josephson Tunnelkontaktes in einem RTM frequenzab-

hängig ist, und des Weiteren, dass die typische Geometrie eines RTM die elektromagnetischen

Eigenschaften einer Monopolantenne teilt, wobei die RTM Spitze der Antenne entspricht.

Der Vergleich der I (VJ)-Spektren von spannungs- mit stromgetriebenen Experimenten ergab,

dass die Zeitentwicklung der Phase in hohem Maße durch die Energiedissipation mittels der

Anregung von Quasiteilchenzuständen beeinflusst wird. Durch die Untersuchung des „Rück-

fallstroms“ IR konnten wir nachweisen, dass (a) sich die Zeitentwicklung der Phase durch

eine klassische Bewegungsgleichung beschreiben lässt, und (b) zwei verschiedene Dissipa-

tionskanäle existieren. Bei kleineren Tunnelwiderstandswerten von RN ≤ 150kΩ erfolgt die

Energiedissipation über Andreev Reflektionen, wohingegen für größere Werte des Tunnelwi-

derstands von RN ≥ 150kΩ die Energdiedissipation von Lebenszeiteffekten der Cooper-Paare

dominiert wird. Darüberhinaus ergab der Vergleich beider Experimente, dass sich der quanten-

mechanische Ursprung der Phase in verschiedenen Phänomenen äußert, welche in starkem

v

Acknowledgements

Maße die experimentellen I (VJ)-Charakteristika unserer Experimente für GN ≈G0 verändern.

In summa wurden in dieser Dissertation die Eigenschaften des Josephson Tunnelkontaktes in

einem STM bei Milli-Kelvin–Temperaturen bestimmt. Daher stellt diese Arbeit notwendige

und grundlegende Schritte dar, welche es uns ermöglichen werden, den Josephson Effekt als

eine vielseitige Sonde auf atomarer Ebene einzusetzen.

Aus dem Englischen übersetzt von Dr. Herbert Jäck

Stichwörter: Josephson Effekt, Rastertunnelmikroskop (RTM), Josephson-Rastertunnelmikroskop

(JRTM), Dynamische Coulomb Blockade, Quasiteilchen Dissipation

vi

RésuméL’objectif de cette thèse est de caractériser les propriétés d’une jonction de Josephson dans

un microscope à effet tunnel (STM) à des températures millikelvin et de mettre en applica-

tion le microscope à effet tunnel Josephson (JSTM) comme sonde polyvalente à l’échelle

atomique. Pour y parvenir, nous examinons les caractéristiques de tunnel I (VJ) de la jonction

de Josephson dans notre microscope à effet tunnel (STM) à la température de référence de

15 mK au moyen d’expériences à courant polarisé et tension polarisée. Nous observons que

dans le régime tunnel GN ¿G0, la jonction de Josephson fonctionne au régime du blocage

dynamique de Coulomb (DCB), pendant lequel l’effet tunnel séquentiel des paires de Cooper

domine le courant de tunnel. L’utilisation de la théorie P (E ) nous permet de modéliser expéri-

mentalement les caractéristiques I (VJ) des expériences à tension polarisée et de déterminer

expérimentalement les valeurs du courant critique de Josephson en accord avec la théorie. De

plus, nous observons une rupture de la théorie P (E) pour les expériences à haute valeurs de

conductance de tunnel GN ≈G0, ce qui pourrait indiquer que l’effet de tunnel cohérent des

paires de Cooper contribue fortement au courant de tunnel dans cette limite. Nous observons

également que la jonction de Josephson dans un microscope à effet tunnel (STM) à tempéra-

tures moins de 100 mK est extrêmement sensible à son environnement électromagnétique ce

qui résulte de sa faible capacité de jonction de l’ordre de quelques femtofarads. En combinant

l’expérience à des simulations numériques, il apparaît que l’environnement immédiat de la

jonction de Josephson dans un microscope à effet tunnel dépend de la fréquence et que de

plus, la géométrie typique d’un microscope à effet tunnel partage les propriétés électroma-

gnétiques d’une antenne monopôle, la pointe du microscope agissant comme l’antenne. En

comparant les courbes I (VJ ) des expériences à courant polarisé et tension polarisée, nous

trouvons que l’évolution temporelle de la phase est fortement affectée par la dissipation dûe à

l’excitation des quasi-particules. À partir de l’étude du courant de retrapping IR, nous pouvons

tout d’abord montrer que l’évolution temporelle de la phase de jonction dans notre micro-

scope à effet tunnel satisfait à l’équation classique du mouvement. Ensuite, nous pouvons

déterminer deux canaux quant à la dissipation d’énergie de la phase de jonction. Pour des

valeurs de résistance de tunnel RN ≤ 150kΩ la jonction dissipe via des réflexions d’Andreev

tandis que pour de plus grandes valeurs de résistance RN ≥ 150kΩ, la dissipation de puissance

est déterminée par la longévité des paires de Cooper. De plus, à partir de la comparaison des

deux expériences, nous pouvons observer que la nature quantique de la jonction se mani-

feste au cours de phénomènes mécaniques quantiques, tels que la phase de l’effet tunnel qui

change fortement les caractéristiques I (VJ) pour GN ≤G0.

vii

Acknowledgements

Pour conclure, dans le cadre de cette thèse, nous caractérisons les propriétés de la jonction de

Josephson dans un microscope à effet tunnel, fonctionnant à des températures millikelvin. En

conséquent, ce travail présente les étapes fondamentales nécessaires qui permettent d’em-

ployer l’effet Josephson comme sonde polyvalente à l’échelle atomique.

Traduit de l’anglais par Thierry Gentès

Key words : Effet Josephson, Microscope à effet tunnel (STM), Microscope à effet tunnel Joseph-

son (JSTM), Blocage dynamique de Coulomb, Dissipation de quasi-particule

viii

ContentsAcknowledgements i

Abstract (English/Deutsch/Français) iii

List of figures xiii

List of tables xvii

Abbreviations xix

1 Introduction 1

2 Fundamental Theory of Tunneling with Superconductors 5

2.1 Microscopic Theory of Superconductivity . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Tunneling of Quasiparticles and Cooper Pairs . . . . . . . . . . . . . . . . . . 9

2.2.1 The Tunneling Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Tunneling of Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Andreev Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Finite Lifetime Effects of Cooper Pairs on the Superconducting Density of

States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.5 Tunneling of Cooper Pairs - The Josephson Effect . . . . . . . . . . . . . . 14

2.3 Dynamics of the Junction Phase φ(t ) . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Thermal Fluctuations of the Junction Phase φ(t ) . . . . . . . . . . . . . . . . . . 20

2.5 Quantum Mechanics of a Josephson junction . . . . . . . . . . . . . . . . . . . . 23

3 Experimental 27

3.1 Scanning Tunneling Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Experimental Requirements for JSTM . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Mechanical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Refrigeration to Millikelvin Temperatures . . . . . . . . . . . . . . . . . . . 31

3.2.3 Electronic Temperature and Energy Resolution . . . . . . . . . . . . . . . 32

3.3 Sample and Tip Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Preparation of Vanadium (001) Single Crystal . . . . . . . . . . . . . . . . 35

3.3.2 Preparation of Vanadium Tips . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Measurement Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

ix

Contents

3.5 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 The Critical Josephson Current in the Dynamical Coulomb Blockade Regime 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 P (E)-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Numerical Computation of the PZ (E ) Distribution for an Arbitrary Impedance

Zt (ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2 Thermal Broadening of the P (E) Distribution due to Thermal Voltage

Noise u & Calculation of the Total P (E) Distribution . . . . . . . . . . . . 48

4.3 The Environmental Impedance of a Typical STM Geometry . . . . . . . . . . . . 49

4.4 Implementation of P (E)-Theory as Fitting Routine . . . . . . . . . . . . . . . . . 51

4.5 Testing the Range of Validity for P (E)-Theory . . . . . . . . . . . . . . . . . . . . 52

4.6 The Josephson Critical Current in the Dynamical Coulomb Blockade Regime . 58

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 The AC Josephson Effect as a GHz Source on the Atomic Scale 63

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Tuning the Tip Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Numerical Simulations of the Environmental Impedance . . . . . . . . . . . . . 68

5.4 Features and Perspectives of the AC Josephson STM . . . . . . . . . . . . . . . . 71

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Phase Dynamics of an Atomic Scale Josephson Junction 75

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Defining the Quantum-Mechanical Regime . . . . . . . . . . . . . . . . . . . . . 77

6.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3.1 Retrapping into the Zero Voltage State . . . . . . . . . . . . . . . . . . . . . 80

6.3.2 Stability of the Zero Voltage State . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Numerical Simulation of the Phase Dynamics . . . . . . . . . . . . . . . . . . . . 87

6.5 Comparison of Current-biased and Voltage-biased Experiments . . . . . . . . . 88

6.6 Analysis of the Voltage-biased Experiments . . . . . . . . . . . . . . . . . . . . . . 90

6.7 Analysis of the Retrapping Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.8 Analysis of the Zero Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Conclusion & Outlook 103

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2.1 JSTM: Probing the Superconducting Ground State . . . . . . . . . . . . . 104

7.2.2 Cooper Pairs and Photons I: The AC Josephson Spectrometer . . . . . . . 105

7.2.3 Cooper Pairs and Photons II: Nonlinear Cooper Pair Tunnel Effects . . . 106

7.2.4 Tuning the Energy Scales: Coherent vs. Sequential Cooper pair Tunneling 107

7.2.5 Studying the Quasiparticle Dissipation of Josephson Junctions . . . . . . 108

x

Contents

Bibliography 111

Curriculum Vitae 123

Publications 125

xi

List of Figures2.1 Illustration of Cooper pair formation . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Quasiparticle dispersion relation and density of states . . . . . . . . . . . . . . . 8

2.3 Illustration of the tunneling effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Example of an experimental quasiparticle density of states . . . . . . . . . . . . 12

2.5 Effect of Andreev reflections on the d I (V )/dV spectrum . . . . . . . . . . . . . . 13

2.6 Illustration of the Andreev reflection process . . . . . . . . . . . . . . . . . . . . . 14

2.7 Finite lifetime effects on the quasiparticle density of states . . . . . . . . . . . . 15

2.8 Classical phase dynamics of a Josephson junction . . . . . . . . . . . . . . . . . . 18

2.9 Tunneling characteristics of an overdamped and an underdamped Josephson

junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10 Illustration of the phase diffusion process . . . . . . . . . . . . . . . . . . . . . . 21

2.11 Influence of phase diffusion on the tunneling characteristics of a Josephson

junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.12 Uncertainty in the junction phase and the charge as a function of charging

energy versus Josephson coupling energy . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Illustration of STM operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Illustration of the laboratory hall and the STM head . . . . . . . . . . . . . . . . 30

3.3 Illustration of the dilution refrigerator concept and the cryostat as well as of the

filtering concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Scanning Tunneling Microscopy topography of the vanadium(001) surface in

two different magnifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Experimental d I /dV spectrum of a vanadium-vacuum-vanadium junction . . 37

3.6 Circuit diagrams for voltage-biased and current-biased experiments . . . . . . 37

4.1 Comparison of Josephson coupling energy, capacitive charging energy and ther-

mal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Typical I (VJ ) tunneling characteristics of a voltage-biased experiment on the

Josephson junction in our STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Illustration of inelastic Cooper pair tunneling in the dynamical Coulomb block-

ade regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Comparison of PZ (E ), which only contains the interaction with an environmen-

tal impedance, and the total probability distribution P (E). . . . . . . . . . . . . . 49

xiii

List of Figures

4.5 Electric properties of a typical STM geometry . . . . . . . . . . . . . . . . . . . . 51

4.6 Example of a P (E)-fit to an experimental Cooper pair tunneling curve from a

voltage-biased experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.7 P (E)-fits to experimental tunneling curves for different tunneling conductance

values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.8 Fit parameter of the P (E)-fits to experimental tunneling curves for different

tunneling conductance values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.9 Experimental tunneling spectra for large values of the tunneling conductance . 56

4.10 Investigating the range of validity for P (E)-theory I . . . . . . . . . . . . . . . . . 57

4.11 Investigating the range of validity for P (E)-theory II . . . . . . . . . . . . . . . . . 57

4.12 Determination of the superconducting order parameter from experiment . . . 59

4.13 Comparison of experimental and theoretical values of the Josephson critical

current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1 Conceptual sketch of the STM tip λ/4-monopole antenna . . . . . . . . . . . . . 63

5.2 Experimental tunneling curves for different STM tip lengths . . . . . . . . . . . 65

5.3 Experimental tunneling curves for different tip holder geometries . . . . . . . . 66

5.4 Lowest eigenmode frequency ν0 of the STM tip as a function of the STM tip length 67

5.5 Cross sections of the STM head geometry used for the numerical simulations . 68

5.6 Scattering parameter and impedance of the STM head, as obtained from the

numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.7 Electric field pattern for the STM geometry, as obtained from the numerical

simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.8 Extracting quantities from experimental tunneling curves for the approximation

of the photon field in the tunneling junction and the resonator excitation lifetime 72

6.1 Comparison of tunneling characteristics from both voltage-biased and current-

biased experiments on the Josephson junction in our STM . . . . . . . . . . . . 75

6.2 Classical and quantum mechanical representation of the junction phase . . . . 76

6.3 Calculated eigenfunctions of the junction phase in the quantum-mechanical

description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Comparison of a calculated ohmic current and a calculated quasiparticle current 81

6.5 Approximation of the dissipation regime for the Josephson junction in our STM 83

6.6 Illustration of phase tunneling processes in different representations . . . . . . 86

6.7 Time evolution of the junction phase as obtained by numerical simulations . . 88

6.8 Tunneling characteristics for different values of the tunneling resistance as ob-

tained from numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.9 Comparison of tunneling characteristics from voltage- and current-biased ex-

periments for different values of the tunneling resistance . . . . . . . . . . . . . 89

6.10 Consistency check for the experimental values of the tunneling resistance . . . 90

6.11 P (E)-fit to a tunneling curve from voltage-biased experiments . . . . . . . . . . 91

6.12 Comparison of experimental and theoretical Josephson critical current values . 92

xiv

List of Figures

6.13 Fit parameter, as obtained from the P (E)-fits, as a function of the tunneling

resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.14 Analysis of the quasiparticle density of states using the extended Maki model . 94

6.15 Comparison of experimental, simulated and fitted retrapping currents . . . . . 96

6.16 Comparison of experimental and calculated retrapping currents using experi-

mental values of the differential in-gap quasiparticle resistance . . . . . . . . . 97

6.17 Comparison of the Josephson critical current and experimental as well as calcu-

lated values of the switching current . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.18 Comparison of experimental and calculated zero voltage resistance values as a

function of the tunneling resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.1 First JSTM results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 Outlook on nonlinear Cooper pair photon interaction . . . . . . . . . . . . . . . 107

7.3 Outlook on the transition from sequential to coherent Cooper pair tunneling . 108

xv

List of Tables3.1 Circuit resistance of voltage biased measurements for different amplification

ranges of the current amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

xvii

AbbreviationsA1 Large surface tip holder

A2 Small surface tip holder

AC Alternating current

α Damping parameter of the STM tip eigenmodes

BCS Bardeen, Cooper and Schrieffer-type

c,c† Normal electron operators

CJ Junction capacitance

c2 Parameter of the P (E)-fit, which models the in-gap quasiparticle current

DC Direct current

∆ Amplitude of the superconducting order parameter

∆E Energy resolution

DCB Dynamical Coulomb blockade

DOS Density of states

e Elementary charge

EC Capacitive charging energy

EJ Josephson coupling energy

ET Thermal energy

m Electron mass

ϕ Complex phase of the superconducting order parameter

φ Josephson junction phase

φ Josephson junction phase operator

fM Modulation frequency of the lock-in amplifier

FIM Finite Integral Method

γ, γ† Quasiparticle operators

Γ Cooper pair lifetime parameter

Γk,l Tunneling rate

G0 = 2e2/h Quantum of Conductance

GN Tunnel conductivity

h Planck’s constant

ħ= h/(2π) Reduced Planck’s constant3He, 4He Helium isotopes

HF High-frequency

ı Imaginary number

xix

List of Tables

I Current

I0 Josephson critical current

IB Bias current output of the current source

IR Retrapping current

IS Switching current

IS,Z Switching current in the Zener model

J (t ) Phase-phase correlation function

JSTM Josephson Scanning Tunneling Microscopy

κ Decay constant ofΨk,l inside a tunnel barrier

κ Short-pass parameter of ZT (ν)

kB Boltzmann’s constant

K(k) Jacobi’s full elliptic integral of first kind

l STM tip length

l0 Electrical lengthening of the STM tip

Λ Order parameter of the Ginzburg-Landau equation

Mk,l Matrix tunnel element

ν Frequency

νn STM tip eigenmodes

n = q/(2e) Number operator of the Cooper pairs, which are stored on the junction

Ψk,l(z) 1D electronic wavefunction

Ψ(φ) Wavefunction of the junction phase in phase space

P (E) Energy-dependent probability of a Cooper pair for an inelastic energy exchange process

PC(E) P (E) distribution for an energy exchange with thermal voltage fluctuations u

Pmax Global maximum of the P (E) distribution

PT Probability for the junction phase to tunnel through a washboard potential well

PZ(E) P (E) distribution for an energy exchange with Z (ν)

Q Junction quality factor

q Electric charge that is stored on the tunnel junction

q Junction charge operator

RCSJ Resistively and capacitively shunted junction

ρN Normal metal DOS

ρS Superconducting DOS

R0 Zero voltage resistance of the DC Josephson current

RDC=Z(0) DC contribution to the environmental impedance Z (ν)

Reff Effective resistance inside the superconducting gap

RN Tunnel resistance

RQ = h/(4e2) Klitzing’s constant for superconductors

RQP Quasiparticle resistance

RQPP Quasiparticle-pair interference resistance

Ref Reference

RF Radio-frequency

σi Standard deviation of the quantity i

xx

List of Tables

S11 Scattering parameter as obtained from FIM simulations

STM Scanning Tunneling Microscopy

T Temperature

TC Critical temperature of a superconductor

Teff Effective electronic temperature

U Washboard potential barrier height

u Thermal voltage noise on C J

UHV Ultra-high vacuum

V Voltage

VB Bias voltage output of the voltage source

VJ Voltage across the tunnel junction

VM Modulation amplitude of the lock-in amplifier

ω0 Josephson plasma frequency

Z (ν) Frequency-dependent environmental impedance

ZT (ν) Frequency-dependent environmental impedance that is shunted by C J

xxi

1 Introduction

In the year of 1962, the Cambridge student Brian D. Josephson theoretically investigated

possible tunneling phenomena in superconductor-insulator-superconductor tunnel junctions.

Remarkably, for his theoretical considerations Josephson threw over with the hypothesis that

superconducting particles, the Cooper pairs, can not cross a tunnel barrier, a hypothesis that

was established from leading theoreticians in the field of superconductivity 1 [2, 3]. In contrast

to the concept given by Bardeen [2, 3], Josephson assumed that the wavefunction of this

ground state behaves similarly to normal electron wavefunctions and exhibits an exponential

decay outside the superconducting material. As a consequence, the wavefunctions of two

superconducting ground states separated by an oxide barrier could also overlap inside the

barrier and enable the tunneling of Cooper pairs, similar to the tunneling of single electrons

that had been experimentally demonstrated just a few years earlier by Ivar Giaever [4, 5].

Based on this idea, Josephson theoretically predicted the “Josephson effect“ which involves

the tunneling of Cooper pairs and manifests in two different phenomena. First, the DC

Josephson effect describes the dissipationless tunneling of Cooper pairs, which allows one to

observe a tunneling current at zero voltage. Second, the AC Josephson effect describes the

Cooper pair transfer across the tunnel barrier via absorbing a photon of energy E = hν and

therefore acts as a perfect voltage VJ to frequency ν converter 2eVJ = hν, where e denotes the

elementary charge and h denotes Planck’s constant.

The underlying mechanism of the Josephson effect has strong connection to the superconduct-

ing ground state, which represents a macroscopic quantum state [1]. In detail, each ground

state can be described by its complex order parameter ∆= |∆|exp(ıϕ) that is of amplitude ∆

– denoting the coupling energy of the electrons in a Cooper pair – and phase ϕ. For disjoint

superconducting reservoirs, e.g. two superconducting electrodes separated by a thin oxide

tunneling barrier 2, the phase valuesϕ of each reservoir are independent so that a phase differ-

1In 1957, Bardeen, Cooper and Schrieffer (BCS) obtained a theoretical model that successfully describes howsuperconductivity develops in a microscopic picture [1]. They showed that an attractive interaction betweenelectrons leads to the formation of a macroscopic ground state characterized by its order parameter ∆. Microscopi-cally, the attractive interaction, mediated by small lattice distortions, promotes the formation of Cooper pairs thatconsist of two electrons of opposite spin and opposite momentum.

2In fact, the Josephson effect can be observed in any sample in which two superconducting reservoirs are

1

Chapter 1. Introduction

ence φ=ϕ1 −ϕ2 between the two reservoirs can exist. Josephson could show that this phase

difference φ of the coupled wavefunctions is the driving force for a Cooper pair tunneling

current I across the barrier [7]

I = I0 sin(φ), I0 = π

2

∆

RN, (1.1)

where the amplitude of this current, I0, called the Josephson critical current, is solely deter-

mined by the superconducting order parameter ∆ and the tunneling resistance RN of the

barrier [8].

Only one year after their prediction in 1962 both the DC and AC Josephson effect were ex-

perimentally observed in tunnel experiments on superconductor insulator superconductor

junctions [9, 10]. Although Josephson’s theoretical work was initially questioned, the exper-

imental proof of the Josephson effect ignited tremendous research on these Cooper pair

tunneling phenomena which ultimately resulted in a variety of applications. These are, for

instance, the SQUID, which is a highly sensitive magnetometer that has been established as

a standard tool in magnetometry [11]. A second application of the Josephson effect is the

so-called Josephson phase qubit which serves as a central building block for the currently

most sophisticated approach to quantum computing [12]. Moreover, the national institutes

of metrology exploit the AC Josephson effect as a voltage standard that precisely defines the

electric potential [13].

Apart from these successful applications the Josephson effect has also high potential to be

employed as a probe on the local scale. The combination of the Josephson effect with Scanning

Tunneling Microscopy (STM), also referred to as Josephson Scanning Tunneling Microscopy

(JSTM) [14], can serve as an ideal probe for studying, for instance, superconductivity. In

detail, one takes advantage of the Cooper pair current being sensitive to both the amplitude

of the superconducting order parameter ∆ as well as to the phase ϕ. These features allow

one to address open questions on the symmetry of the superconducting order parameter in

non-conventional superconductors, such as Cuprates [14, 15]. JSTM represents also an ideal

tool to study how the superconducting state responds to perturbations that can be introduced

by a single magnetic atom on top of a superconducting surface, for instance [16]. For such

a system, theory predicts a strong response of the superconducting state, which manifests

itself in variations of ∆, including even changes of sign [17, 18]. However, in experiments

on such systems no change in the amplitude of ∆ was observed so far [16, 19, 20]. This is

most likely due to the reason that standard tunnel experiments probe the energy of so-called

quasiparticles that result from breaking up a Cooper pair [4, 5]. Therefore, these experiments

are not a direct probe of the order parameter∆ itself. Here, JSTM provides clear advantages and

could help to shed more light on the fundamental interaction between superconductivity and

magnetism [20], a question that is nowadays also highly discussed in the context of iron-based

superconductors [21].

connected via a weak link. The term weak link can be realized via a short constriction, a normal metallic film orjust an insulating film [6].

2

The AC Josephson effect also has large potential to be employed in JSTM experiments, since it

perfectly converts voltages to frequencies. It was shown in experiments on planar Josephson

junctions that the AC Josephson effect can be employed to sense hyperfine transitions in

rare earth atoms inside the tunnel barrier or to prove the existence of the Andreev bound

state in superconducting atomic point contacts [22, 23]. This spectroscopic feature of the

AC Josephson spectrometer is certainly also of interest in STM experiments. Here, it could

be employed to investigate the spin state of molecular magnets [24], which are a promising

candidate for molecular electronics, or even probe the hyperfine transitions of single atoms

carrying a nuclear spin [25].

The diversity of the given examples illustrates that the combination of the Josephson effect

and STM represents a promising tool in solid state physics and surface science. For this

reason, JSTM already attracted some experimental interest [26, 27, 28, 29, 30], yet it never

experienced a real breakthrough. The reason for the limited success may be found in the very

small coupling strength between the two superconducting wave functions when compared

to other relevant energy scales in experiments, such as thermal energy ET = kBT (kB denotes

Boltzmann’s constant and T the temperature). The condition for Josephson STM to work, as

given in Ref. [14], requires that the Josephson coupling energy EJ exceeds thermal energy ET

such that the coupling between the superconductors is not perturbed by thermal fluctuations

[31]. For STM experiments with elemental superconductors EJ is typically smaller than 20µeV,

which requires experimental temperatures of T ≤ 250mK. All attempts on JSTM so far [26, 27,

28, 29, 30] have been performed at much higher temperatures so that the condition for JSTM

to work were clearly violated. Since recently these necessary very low temperatures are also

accessible to STM setups [32, 33, 34, 35, 36] so that it would now be appropriate to say: JSTM

is in the starting blocks.

In this sense the focus of this work will be to bring JSTM from the starting blocks onto the race

track, i.e. to fully characterize the properties of a Josephson junction in an STM. To this end,

we perform JSTM experiments on the first ultra-high vacuum STM setup capable of reaching

an effective electronic temperature far below 100 mK, which is located in the department of

Prof. Klaus Kern at the Max-Planck-Institute for Solid Research in Stuttgart, Germany [35].

Of particular interest in this work is the question of how the small STM junction capacitance

effects the properties of the Josephson junction. Whereas for large planar junction geometries

the capacitive charging energy EC is negligible [9], the charging energy for an STM junction

capacitance of a few femtofarads is of similar magnitude as the Josephson coupling energy. In

this situation, the Josephson junction interacts with its immediate environment such that the

inelastic tunneling of Cooper pairs can be observed at finite voltages, which is accompanied by

photon emission [37, 38, 39]. Additionally, the phase of this type of Josephson junction exhibits

strong quantum properties that drastically change the junction characteristics [31, 40, 41]. In

fact, the tunneling characteristics of the Josephson junction in our STM are significantly ef-

fected by quantum-mechanical phenomena and require a detailed investigation. In particular,

the determination of experimental values of the Josephson critical current as a probe for the

local superconducting order parameter demands an appropriate theoretical description of

3

Chapter 1. Introduction

the experimental data in view of the successful implementation of JSTM.

At this point it should be noted that JSTM also allows one to continuously tune the Josephson

coupling energy by changing the tip-sample distance. In this way, JSTM can explore new

tunneling regimes and dissipative interactions of the junction with its environment that

are inaccessible to experiments on planar junction geometries. In these experiments EJ

can not be tuned that easily. The tunability of EJ versus EC is of particular interest in the

context of Hamiltonian by design to optimize the properties of phase qubits for quantum

computing [12]. Hence, JSTM also represents ideal means to address fundamental questions

of superconducting tunneling, which extends and complements the diverse portfolio of this

highly interesting technique.

The outline of this thesis is the following: The focus of Chapter 2 is to provide a theoretical

background on superconductivity as well as on the tunneling phenomena of single particles

and Cooper pairs. In addition, some general aspects of the Josephson effect are presented that

help the reader to obtain a fundamental understanding of this effect, whereas detailed theory

is given in the corresponding chapters where needed. Chapter 3 is focused on experimental

aspects of this study, such as the functionality of an STM, refrigeration using a dilution refrig-

erator and aspects on the sample preparation and measurement procedures. In Chapter 4, we

investigate the tunneling characteristics of voltage-biased Josephson junctions and implement

theory that allows us to extract the Josephson critical current from the experimental data.

In Chapter 5, we show that the atomic scale Josephson junction in our STM is sensitive to

high-frequency signals in the GHz range and, moreover, we demonstrate the realization of the

AC Josephson spectrometer in an STM. In Chapter 6, we investigate the dynamics of the phase

difference φ and its dissipative interaction with quasiparticle excitation processes. Moreover,

we compare the experimental results from current-biased and voltage-biased experiments

in order to understand how significantly quantum mechanics effects the properties of the

Josephson junction. Chapter 7, summarizes all results obtained in this work and provides

an outlook on perspective experiments as well as first results of JSTM on the interaction of a

single magnetic impurity with a superconducting ground state.

4

2 Fundamental Theory of Tunnelingwith Superconductors

The goal of this chapter is to provide the reader with theoretical basics of tunneling between

superconducting electrodes. These involve the microscopic description of superconductivity

by Bardeen, Cooper and Schrieffer as well as general aspects of the tunnel effect. Second, this

chapter includes the derivation of the Josephson equations as well as their generalization and

describes fundamental physical aspects of Josephson junctions. The chapter closes with a

discussion of the influence of perturbations on the Josephson effect. Further theory that is

needed for particular experiments will be considered in the corresponding chapters.

2.1 Microscopic Theory of Superconductivity

The foundation of the modern understanding of superconductivity and related phenomena

was set when J. Bardeen, L. N. Cooper and J. R. Schrieffer (BCS) obtained their microscopic

description of superconductivity, the BCS theory [1]. This theory succeeded in connecting a

quantum-mechanical picture of particle interactions with the critical temperature TC of the

phase transition, at which a normal metal turns superconducting. The fundamental concept

behind this theory is an attractive interaction between electrons in a metal mediated by the

crystal lattice. A first indication for this interaction potential was experimentally provided

by the so-called isotope effect. It was found that the critical temperature TC depended on

the isotope of the specific element, thus on the ion mass. This observation pointed to a

significant involvement of the crystal lattice for the formation of the superconducting phase.

Based on these experimental results, Fröhlich and Bardeen were able to provide a physical

picture as well as a theoretical description for an attractive interaction between electrons

[42, 43]: In a simplified picture, when the negatively charged electron moves through the

crystal, it will attract positively charged ions sitting on the lattice sites when passing by. This

lattice distortion creates a local, positive charge density that creates an attractive, albeit small,

potential between a pair of electrons, as shown in Fig. 2.1. Bardeen, Cooper and Schrieffer

found that this pair of electrons is formed by two electrons of opposite momentum ±k and

opposite spin Sz =±1/2. This particle, called a Cooper pair, is of bosonic nature with a total

spin S = 0 [1].

5

Chapter 2. Fundamental Theory of Tunneling with Superconductors

Figure 2.1 – Passing through the crystal lattice of ions (orange circles), the electrons (bluecircles) induce a small lattice distortion. This distortion yields a positive, localized spacecharge n+

l , that results in a attractive potential between two electrons of opposite momentum,±k, and opposite spin, ↑,↓.

To derive an expression for the order parameter of the superconducting phase it is necessary

to consider the Hamiltonian of an electronic system that includes this electron-electron

interaction potential ζ. For the sake of simplicity, this potential is chosen to be momentum

independent. In the framework of BCS theory, this Hamiltonian in second quantization and

momentum space reads as:

H −µN = ∑k,σε(k−µ)c†

k,σckσ+ζ∑k,k′

c†k↑c†

−k↓ck′↓c−k′↑+h.c.. (2.1)

The first term on the right side of the equation, ε(k) =ħk2/2m, denotes the kinetic energy of a

free electron with respect to the chemical potential µ of a reservoir of N electrons. The particle

operators c and c† anihilate an electron or create an electron of spin state σ, respectively.

The second term represents the superconducting ground state,∑

k′ ck′↓c−k′↑, and its complex

conjugate, that result from the attractive interaction ζ. In a more descriptive explanation

this last term sums up the entirety of Cooper pairs in the superconducting phase. Since the

attractive potential was chosen to be momentum independent, a mean-field approach on the

superconducting ground state yields

HMF −µN =∑k

(c†

k↑c−k↓)(ε(k)−µ ∆

∆∗ µ−ε(k)

)(ck↑

c†−k↓

)− ∆

∗∆ζ

. (2.2)

This equation already contains the order parameter of the superconducting phase ∆ = ζ×∑k⟨ck′↓c−k′↑⟩ and its complex conjugate. To calculate the eigenvalue spectrum of the super-

conducting ground state it is necessary to diagonalize Eqn. 2.2 by rotating the operator basis.

6

2.1. Microscopic Theory of Superconductivity

This is obtained by the Bogoliubov-transformation, which introduces novel operators

γ1,k = uck↑+ vc†−k↓, (2.3)

γ2,k = −v∗ck↑+u∗c†−k↓, (2.4)

where u and v are complex numbers obeying the condition |u|2 +|v |2 = 1. Within this new

basis the mean-field BCS Hamiltonian is diagonalized and reads as

HMF −µN =∑k

(γ†

1,kγ†2,k

)×

(Ek 0

0 −Ek

)(γ1,k

γ2,k

)− ∆

∗∆ζ

. (2.5)

Ek = ±√

(ε(k)−µ)2 +|∆|2 represents the eigenvalue spectrum of the Hamiltonian, whose

momentum dependence, the dispersion relation, is shown in Fig. 2.2(a). It features an energy

gap that has twice the width of the order parameter ∆ around the Fermi energy EF =µN and a

parabolic dispersion to higher and lower energies, respectively. The underlying phenomenon

is that the electrons condense at the Fermi energy to form the superconducting ground state∑k ck↓c−k↑. If the energy E = 2∆ is provided, quasiparticles of this ground state can be excited

by the operator γ† having the energy Ek. For this reason the γ operators are also called

quasiparticle operators. A more descriptive explanation is obtained in the single particle

picture, where electrons form Cooper pairs of zero kinetic and angular momentum energy

at the Fermi energy EF . If the energy E = 2∆ is provided, a Cooper pair is broken apart and

a single electron is generated. This will now behave as a free electron in a metal for which

reason it follows its quadratic dispersion relation.

From the dispersion relation Ek, it is straightforward to calculate the density of states ρ of the

superconductor

ρS = d N (E)

dE= d N (ε)

dε

dε

dE= ρN

EpE 2 −∆2

, (2.6)

with ρN as the normal metal density of states (DOS). ρS is shown in Fig. 2.2(b) and features a

singularity at E =∆ and a gap of width 2∆0. Its energy dependence is evident when we sum up

all states, equally spaced in k, for all energies Ek of the dispersion relation in Fig. 2.1(a).

To determine the order parameter of the superconducting ground state ∆ in BCS theory, it is

necessary to perform a self-consistent calculation starting from its definition [1]:

∆= ζ×∑k⟨ck↓c−k↑⟩ =V

∑k

[u∗v⟨γ†

1,kγ1,k⟩−u∗v⟨γ†2,kγ2,k⟩

]. (2.7)

Including a fermionic distribution of quasiparticle states ⟨γ†γ⟩ = 1/(exp(Ek/kB T )+1), and

replacing the sum over all states in momentum space by an energy integral on the normal

7


Figure 2.2 – (a) One-dimensional quasiparticle dispersion relation Ek normalized to ∆ as afunction of momentum k for EF = 0. (b) Quasiparticle density of states ρS normalized to themetallic density of states ρN as a function of energy E .

density of states ρN , some algebra yields:

1 = |ζ|2

∫ρN dε

1√(ε(k)−µ)2 +|∆|2

tanh

(√(ε(k)−µ)2 +|∆|2

2kB T

). (2.8)

For temperatures larger than the critical temperature T > TC , the only solution possible is

∆= 0, which is a reasonable result since at this temperature no superconductivity is expected.

In the opposite case, T < TC , and assuming that the energy interval µ± εD of interaction

around the Fermi level µ is determined by the Debye energy εD – a measure for the excitation

energy of phonons – the self-consistent equation for ∆ yields a finite value for ∆. Assuming

weak electron-phonon coupling an expression for the order parameter in the zero temperature

limit ∆0 can be derived [1]:

∆0 = 1.764kB TC . (2.9)

This relation between∆0 and TC is of universal validity for all weakly-coupled superconductors

and has been experimentally proven for numerous materials.

Although the superconducting order parameter∆ could be derived from a microscopic picture,

other important properties were first discovered when Lev Gorkov succeeded in connecting

the phenomenological Ginsburg-Landau theory with the microscopic BCS-theory [44]. The

Ginsburg-Landau theory is a thermodynamical approach to superconductivity that relates

the superconductor’s free energy to a complex order parameter of the phase transition Λ

[45]. Gorkov found that the microscopically derived order parameter ∆ is proportional toΛ.

Accordingly, also∆=∆e iΦ is complex with a phaseΦ, which induces significant consequences

as we will see in the next section. Moreover, Gorkov could identifyΛ as the wavefunction of

Cooper pairs and show that Cooper pairs have an electric charge q = 2e.

8

2.2. The Tunneling of Quasiparticles and Cooper Pairs

2.2 The Tunneling of Quasiparticles and Cooper Pairs

To evaluate an expression for the tunneling current that involves superconducting tunnel-

ing electrodes, this section will first deal with general aspects of the tunneling effect. This

introductory part is followed by theoretical considerations for quasiparticle tunneling, before

the Josephson effect itself is introduced. It should be further noted that the following theory

is presented in chronological order and mirrors the remarkable progress of theoretical and

experimental knowledge on electronic tunneling between the years 1959 and 1961.

2.2.1 The Tunneling Effect

The tunneling effect is a manifestation of the fundamental concept in quantum mechanics

where a particle is described by a probability wavefunctionΨ(x,k). This wavefunction rep-

resents the probability to find the particle at location x having the momentum k = p/ħ. If

such a quantum-mechanical particle, an electron for instance, is located nearby a barrier of

finite width and finite height, its wavefunction will not immediately stop at the barrier, but

penetrate through and decay behind it as illustrated in Fig. 2.3(a). To a certain probability

the electron can therefore be found behind the barrier; it has tunneled through the barrier.

Experimentally, this tunnel effect was first observed in 1897 by Robert W. Wood investigating

the field emission of electrons into vacuum. However, he was not able to describe it, since,

at this time, the quantum-mechanical concept was unknown. Twenty years later F. Hund

succeeded in explaining the tunnel effect, observed in experiments on molecular isomers, in a

quantum-mechanical fashion before, one year later, G. Gamow could derive a mathematical

description for this effect starting from the quantum-mechanical Schrödinger equation [46].

The distance behind the barrier at which the tunnel effect can still be observed, relates to

the properties of the barrier and the wavefunction. Considering, for instance, the interface

of a bulk piece of metal and vacuum as a barrier for the electronic wavefunction Ψ inside

the metal, its extension into the vacuum will be on the order of nanometers. Accordingly,

when two pieces of metal are separated by such a small distance the wavefunctions of both

metals will overlap, as shown in Fig. 2.3(b). This situation is called tunnel contact and offers a

possibility to measure the quantum-mechanical tunnel effect using electrons: If an external

voltage V is applied to such a contact, electrons can tunnel from an occupied electronic state

Ψk in one electrode into an unoccupied stateΨl in the other one and therefore, a tunneling

current can be measured.

2.2.2 Tunneling of Quasiparticles

First tunneling experiments were performed on devices having one superconducting and

one metallic electrode, which are separated by a thin insulating oxide layer [47, 4, 5]. The

experimental results on the measured tunneling current I (V ) confirmed that it originates

from an overlap of wavefunctions in the tunnel barrier. It was further observed that the

9


Figure 2.3 – (a) Classical and quantum-mechanical representation of a particle: The red dotis either on the right or the left side of the barrier, showing classical behavior. In contrast, aquantum-mechanical particle represented by a wavefunction has – as a result of the tunnelingeffect – a probability to be found on both sides. (b) Tunneling of an electron between twoelectrodes: The wavefunctionsΨk (z) andΨl (z), representing single electrons, overlap in thetunnel barrier of height U . If a voltage V is applied, electrons can tunnel from an occupiedstate in one electrode to an unoccupied state in the other electrode.

measured differential conductance d I /dV directly relates to the density of states (DOS) of the

superconductor ρl (E ), for which reason the tunneling current has to be related to quasiparticle

tunneling [47, 4, 5]. Interestingly, these first measurements on electron tunneling were already

performed on devices involving superconducting electrodes. This could be possibly explained

by the fact that the superconducting DOS has strong features compared to DOS of a normal

metal.

Stimulated by these experimental studies, Bardeen could draft a first theoretical picture for

the electronic tunneling process: The starting point is the assumption that the electronic

wavefunctions of both electrodes, including the superconductor, are free electron plane

wavefunctions Ψk,l , which exponentially decay inside the potential barrier (z0 < z < z1) of

height U , as shown in Fig. 2.3(b). These wavefunctions, where z corresponds to the direction

normal to the interface S in the x y-plane, read as:

Ψ1,k,l (z) ∝ sin(kz z) (2.10)

Ψ2,k,l (z) ∝ exp(κz), z0 < z < z1. (2.11)

Here, κ=p

2mU /ħ2 denotes the decay constant of the wavefunction inside the barrier. In-

cluding these wavefunctions in time-dependent perturbation theory, Bardeen could derive

expressions for both the tunneling matrix element Mk,l – containing the overlapping wavefunc-

tions across the entire tunnel contact surface area S – and the energy-dependent tunneling

10


rate Γk,l (E) [2]:

Mk,l = − ı

2m

∫dSΨ∗

2,k

∂Ψ2,l

∂z−Ψ2,l

∂Ψ∗2,k

∂z, (2.12)

Γk,l (E) = (2π/ħ)|Mk,l |2ρl (E) fk (E)(1− fl (E)). (2.13)

With this result it was, for the first time, possible to qualitatively describe an experimentally

determined tunneling spectrum between a normal metallic and a superconducting electrode.

However, Bardeen argued, referring to Gorkov’s work, that the superconducting order parame-

ter drops to zero rapidly inside the barrier so that a paired wavefunction cannot exist in this

region [2]. Accordingly, the sole difference to tunneling experiments with normal metallic

electrodes arises from the superconducting DOS [2, 3], a conclusion that clearly was proven

incorrect later on.

A more general picture on the tunneling between a metallic and a superconducting electrode

was derived one year later by Marvin H. Cohen et al., as an extension to Bardeen’s work in a

manybody calculation [48]. The starting point of this calculation is the Hamiltonian of that

system including a term for interaction,

H = HN +HS +HT . (2.14)

It contains the normal conducting electrode HN that corresponds to the normal conducting

part of Eq. 2.1, the superconducting electrode HS , which in a similar fashion contains the

number operators NS = ⟨γ†γ⟩ of the quasiparticle operators (see Eq. 2.3), and an interaction

Hamiltonian HT :

HT = ∑k,l ,σ

[Mk,l c†k,σcl ,σ+Ml ,k c†

l ,σckσ]. (2.15)

As in Sec. 2.1 the operators c,c† denote single electron particle operators and the matrix

element Mk,l is similar to Eq. 2.12, denoting the overlap of normal electronic wavefunctions in

the barrier (similar to Bardeen’s approach). In this picture the tunneling rate corresponds to

the expectation value of the change in quasiparticles in the superconducting electrode < NS >.

Starting from the equation of motion of the quasiparticle number operator ıħNS = [NS , HT ],

Cohen et al. succeeded in deriving an expression for the tunneling rate Γ that depends on the

applied voltage V [48]:

Γ(V ) = ⟨NS(V )⟩∝∫ ∞

−∞|M |2[ fN (E ′−eV )− fS(E ′)]ρN (E ′−eV )ρS(E ′)dE ′. (2.16)

This equation reveals that the tunneling rate of a superconductor-insulator-metal junction

corresponds to the convolution of the superconducting DOS ρS (see Eq. 2.6) with the normal

conducting DOS ρN . In good approximation, ρN is constant around the Fermi energy on an

interval relevant for superconductivity, ∆E ≈ 1meV (this assumption holds for all elemental

11


superconductors). For this reason, an experimentally determined differential conductance

spectrum d I (V )/dV , as shown Fig. 2.4, will be similar to the theoretical superconducting

DOS ρS (cf. Fig. 2.2(b)). Yet, deviations from an ideal BCS gap, as introduced in Sec. 2.1, can

be observed such as a finite width in the quasiparticle excitation peaks, as indicated by the

arrows. These deviations can originate from thermally excited quasiparticles, lifetime effects

of Cooper pairs and so called Andreev-reflections. For the experiments performed in this work

at a temperature of T = 15mK (cf. Ch. 3), thermally excited quasiparticles can be neglected,

since thermal energy kB T ≈ 3.5µeV is negligibly small compared to typical BCS gaps on the

order of ∆À 100µeV. The latter two effects will be addressed in the following two subsections.

Figure 2.4 – Experimental d I (V )/dV spectrum as a function of the applied voltage bias Vmeasured at a temperature T = 15mK with the voltage axis normalized to the superconduct-ing order parameter ∆. As electrode material we used vanadium, which is a typical BCSsuperconductor, as the STM tip and a Cu(100) surface as the sample.

2.2.3 Andreev Reflections

In the last section it was shown that the differential tunneling conductance d I (V )/dV of a

superconductor-insulator-metal junction is similar to the superconducting DOS as given in

Eq. 2.6 and Fig. 2.2. Hence, the d I (V )/dV spectrum should only feature the sharp coherence

peaks symmetrically centered around the Fermi energy and an empty gap region. However, a

filling of the gap could also be observed experimentally when the tunnel contact resistance RN

was drastically reduced to smaller values. These in-gap features originate from pair creation

processes via quasiparticle tunneling and were predicted by Andreev just two years after

Cohen’s work on superconducting tunneling [49]. For this reason these processes are called

Andreev reflections. These reflections involve an electron in the metallic electrode that is

reflected by the tunnel barrier as a hole. In order to conserve charge and energy, this process

creates a Cooper pair in the superconducting electrode on the other side of the barrier upon

12


reflection of the hole 1. Since this process involves the transfer of two particles across the

tunnel barrier the tunneling probability of this process depends on the tunneling resistance as

1/R2N .

Tinkham et al. [50] succeeded in deriving a theoretical model that relates the closing of the gap

to a so-called barrier parameter Z . Although it is difficult to relate this barrier parameter to the

real tunnel contact resistance Fig. 2.5 already gives an impression of this gap-filling effect. For a

vanishing tunnel barrier, Z = 0, the electrodes are in metallic contact and the superconducting

gap is filled up, exhibiting a table-like plateau. In the opposite case of a strong barrier, Z = 10,

the gap remains open. This latter situation corresponds to the experimental situation of a

tunnel contact, i.e. RN À 1/G0.

Figure 2.5 – Calculated differential tunneling conductance d I (V )/dV as a function of thebarrier parameter Z [50].

Replacing the metallic electrode by a second superconducting electrode with the same gap ∆

enables the observation of further Andreev reflections inside the gap [51, 52]. If the voltage

drop across the tunnel junction corresponds to the gap value, eV = ∆, an electron can be

reflected as a hole and a Cooper pair is created, as illustrated by Fig. 2.6. This process manifests

itself as a defined peak inside the superconducting gap at eV = ∆. If the two gaps are of

different value it is even possible to observe multiple Andreev reflections (MAR), which involves

the transfer of more than two quasiparticles. These processes yield in-gap structures at

eV = (∆1 +∆2)/n. However, their experimental observation is more challenging due to the

reduced tunneling probability, which is proportional to 1/RnN . A nice experimental work on

these MAR structures, which was also performed using an STM, is given in Ref. [53].

1Certainly, this process also occurs in the oppposite direction via the annihilation of a Cooper pair.

13


Figure 2.6 – Illustration of the two-particle Andreev reflection that generates a Cooper pair atvoltage V =∆ [51].

2.2.4 Finite Lifetime Effects of Cooper Pairs on the Superconducting Density ofStates

In many experiments the observed superconducting DOS deviates from the perfect case

displayed in Fig. 2.2. The coherence peaks do not appear perfectly sharp but instead are

broadened and are of finite width. The origin of this broadening can be manyfold, such as

a finite lifetime of Cooper pairs due to scattering at surfaces or spin-orbit effects in heavy

superconductors. Yet, this experimentally observed broadening can be adressed phenomeno-

logically by introducing a so-called lifetime parameter Γ, as proposed by Dynes et al. [54]. This

lifetime parameter is simply added to the superconducting DOS in Eq. 2.6, which modifies to

ρ′S(E) = ρNℜ

[E − ıΓ√

(E − ıΓ)2 −∆2

]. (2.17)

Figure 2.7 displays the modified differential tunneling conductance d I (V )/dV for different

values of Γ. It can be clearly seen that Γ not only broadens the quasiparticle excitation peaks

but also reduces their amplitude.

2.2.5 Tunneling of Cooper Pairs - The Josephson Effect

In the same year that Cohen et al. developed their theory on tunneling between a metal and

a superconductor, Brian D. Josephon was theoretically investigating a tunnel junction that

consists of two superconducting electrodes, which the indices k and l separated by a thin

oxide layer as tunneling barrier. Josephson’s fundamental idea was that despite the fact the

14


Figure 2.7 – Calculated differential tunneling conductance d I (V )/dV normalized to the super-conducting gap as a function of the lifetime parameter Γ using Eq. 2.17.

order parameter Λ rapidly drops to zero inside the tunnel barrier, the pair wavefunctions

of both superconductors may still overlap. Moreover, he argued that two superconducting

ground states, ∆k and ∆l , separated by a tunnel barrier, should also have independent phases

Φk 6=Φl , corresponding to a different number of Cooper pairs in the electrodes. From these

assumptions, Josephson concluded that the coherent superposition of these two ground

states of different phases φ=Φk −Φl should enable the creation of a ground state pair in one

electrode Sk † and the annihilation of such a pair in the other one Sl , whereas the total number

of single particle states N remains unchanged, Sk † Sl |N ,φ>= e ıφ|N ,φ>.

Including these assumptions in the system’s Hamiltonian, which is similar to Eq. 2.14, Joseph-

son followed Cohen’s perturbative approach in calculating the average rate of change in the

quasiparticle density of one electrode < Nl > and found the following expression for the total

current through the junction:

I (t ) = IQP + 1

2I0Sl (t )†Sk (t )+ 1

2I∗0 Sk (t )†Sl (t ) (2.18)

The first term, IQP , represents the quasiparticle current, which is similiar to Cohen’s expres-

sion in Eq. 2.16. The last two terms represent a pair transfer current of amplitude I0, which

oscillates at a frequency of dφd t = 2eVJ /h, where VJ is the voltage drop across the tunnel contact.

Although the theoretical description Josephson found was quite general, he was already able

to interpret the physical significance of the pair current terms and predicted two phenomena,

the Josephson effects [7]:

1. The AC Josephson effect: A ground state pair transfer at the voltage 2eVJ via the absorp-

tion or emission of a photon having the frequency ν= 2eVJ /h.

2. The DC Josephson effect: A ground state pair transfer at zero voltage,VJ = 0, having the

15


maximum amplitude I0, which is called the Josephson critical current.

The physical interpretation of these two effects is that, first, in a Josephson junction a tunneling

current of a maximum amplitude I0 can be observed at zero voltage. Second, a tunneling

current can be measured at voltage VJ if a photon of frequency ν= 2eVJ /h has been absorbed

or emitted by the junction 2.

Josephson’s derivations were quite revolutionary at this time since they were in contradiction

to the established hypothesis that paired wavefunctions can not exist inside the tunnel barrier

[2]. Even half a year after Josephon published his theory, theoretical studies appeared referring

to this hypothesis [3]. Nevertheless, it was Josephson’s mentor himself, P. W Anderson, who

could – together with J. Rowell – experimentally demonstrate the existence of the DC Josephson

effect [9], shortly before S. Shapiro also observed the AC Josephson effect [10]. One year after

Josephson’s inital work, V. Ambegaokar and A. Baratoff could not only theoretically validate

his theory, but also significantly extend it [8, 44]. Following the same perturbative approach

for calculating < Nl >, but representing the particle operators by their Green’s functions, they

derived the famous expression for the dependence of the pair current on the phase-relation

between the two superconducting ground states:

I (φ) = I0 sin(φ). (2.19)

Second, they also determined the coupling energy E(φ) of the two superconducting ground

states as well as the temperature-dependent critical current I0(T ) of the pair current, yielding:

E(φ) = E J cos(φ) =− ħ2e

I0 cos(φ), (2.20)

I0 = ∆1(T )

RNK

(√1−∆2

1/∆22

). (2.21)

Here, E J denotes the Josephson coupling energy of the junction that relates to the critical

current as E J = (ħ/2e)I0. ∆ denotes the modulus of the superconducting order parameter,

of which ∆1 is the smaller one, RN the resistance of the tunnel contact in the metallic state,

and K Jacobi’s complete elliptic integral of first kind. The physical interpretation of these

two equations is that the overlap of the pair wavefunctions in the tunnel barrier results in a

coupling potential E . With respect to the phase, E forms a so-called washboard-potential, in

whose minima a particle – representing the junction – is in a bound state at constant φ, as

displayed in Fig. 2.8(a). The height of the potential walls, E J , is solely determined by ∆ and

RN . The tunneling resistance of the Josephson junction RN ∝ Mk,l directly reflects the pair

wavefunction overlap via the matrix element Mk,l (compare Eq. 2.12). Moreover, it should

be noted that Eq. 2.21 for the critical current only holds for weakly coupled electrodes since

the pair transfer was evaluated as a perturbation to the whole system. In this context, weakly

2Observing photon emission from Josephson junctions is not as straightforward as anticipated in early dis-courses, but this aspect will be addressed later.

16

2.3. Dynamics of the Junction Phase φ(t )

coupled refers to a low transmission GN = 1/RN of the junction with respect to the quantum of

conductance G0, such that GN ¿G0.

The above presented, initial studies on the Josephson effect could already predict and prove

its existence. However, they only marked the starting point for the investigation of pair trans-

fer phenomena in tunnel junctions with two superconducting electrodes, called Josephson

junctions. Other important aspects such as the temporal properties of this pair transfer were

addressed afterwards and will be part of the next section.

2.3 Dynamics of the Junction Phase φ(t )

The dynamics of the junction phase are of fundamental importance, since they determine the

experimental characteristics of a Josephson junction. For this reason, right after the prediction

and observation of the Josephson effect, subsequent theoretical investigations focused on

the time-dependence of φ(t). It was Josephson itself who rewrote his first expression of the

tunneling current (Eq. 2.18) such that it connects φ(t ) directly with the current I (t ) through

the junction [55]:

I (t ) = I0 sin(φ(t ))+ [1

RQP+ 1

RQPPcos(φ(t ))]

h

2eφ(t ) (2.22)

Here, the first term denotes the current associated with the DC Josephson effect at zero voltage,

the second term is the dissipative quasiparticle current at finite voltage and the last term is

the so-called quasiparticle interference term. This term involves the creation (annihilation)

of ground state pairs via tunneling of quasiparticles and, therefore, represents a different

formulation for the phenomenon of Andreev reflections (cf. Sec. 2.2.3). Nevertheless, the

temporal dynamics of φ(t) are dominated by the first two terms, whereas the quasiparticle-

interference term is, for most of the tunneling resistance range, small if not zero or similar.

It should be noted that a theoretical treatment of the individual current amplitudes based on

a Green’s function approach was first given by Werthamer [56]. A very descriptive and general

overview on these current contributions is given in Ref. [6], for instance.

The dynamics of φ(t) can be analyzed when considering the simplest case of a Josephson

junction that is biased by a current source IB . Figure 2.8(b) displays the circuit diagram of such

a junction, which also includes the junction capacitance C J , acting as a shunt. The current

through this capacitance reads as IC (t) = Q = C J ×dV /d t . Using the Josephson relation

V = (ħ/2e)dφ/d t , IC (t) can be inserted into Eq. 2.22 in accordance to Kirchoff’s law, which

yields a differential equation for φ(t ) [57, 58]:

I = I0 sin(φ(t ))+ 1

RQP

h

2eφ(t )+C J

ħ2eφ(t ). (2.23)

To examine the dynamics of φ(t), it is a sufficient approximation to replace the dissipative

17


quasiparticle conductance 1/RQP by an ohmic resistor R. Within this so-called resistively and

capacitively shunted junction model (RCSJ) the differential equation for φ(t ) can be rewritten

to [59, 60]

φ(t )+ 1

τφ(t )+ω2

0 sin(φ(t )) = ω20I

I0, (2.24)

τ = RC J , (2.25)

ω0 =√

2eI0

C Jħ, (2.26)

This equation of φ(t ) represents a non-linear, inhomogeneous differential equation of second

order that cannot be solved analytically, yet it corresponds to the classical equation of motion

for a driven rotating pendulum. In analogy, a pendulum with mass m =C J has the coordinate

φ and experiences dissipation expressed in a quality factor Q = τω0. Therefore, the dynamics

of φ can be qualitatively well explained in this descriptive analogue.

Figure 2.8 – (a) Coupling potential of a Josephson junction for the case of no external driveI = 0. The junction is represented by the blue particle. (b) Circuit diagram of a current-biasedJosephson junction. I denotes the current source, RQP the dissipative quasiparticle resistance,C J the junction capacitance and I0 sin(φ) the Josephson junction. (c) Potential of the currentbiased Josephson junction I > I0. (d) For underdamped junctions the particle continues itsmovement down the potential landscape when I < I0.

18

2.3. Dynamics of the Junction Phase φ(t )

To investigate the dynamics of the junction phaseφ(t ) let us first consider a Josephson junction,

of which the external current drive is zero. In this case the time average of φ(t ) is constant in

time and the particle, representing the junction, oscillates around its zero point of motion

inside the washboard potential minimum, as displayed in Fig. 2.8(a). For an increasing external

current drive, 0 < IB < I0 this situation remains stable. However, as soon as I = I0 the potential

has no minimum anymore and the particle will start rolling down the potential landscape

with velocity φ(t), which is displayed in Fig. 2.8(c). This situation corresponds to the case

in which the junction switches out of its dissipationless zero voltage state to finite voltage

V = (ħ/2e)φ. In the classical analogue, the pendulum will oscillate around its zero point

of motion with increasing amplitude, until the external drive is large enough that it swings

over the pinnacle and starts rotating. Moreover, in this running state at IB > I0, the junction

experiences dissipation as a result of the ohmic shunt R, which causes friction to the moving

particle, thus slowing it down. When again lowering the drive I such that IB ≤ I0, as shown in

Fig. 2.8(c), the potential landscape remains tilted but has recovered its minima. Under this

condition the dynamic reaction of the moving particle, i.e. the dynamics of the junction phase,

crucially depends on the slope of the washboard, as given by the current drive and the amount

of friction, i.e. energy dissipation, which is measured by the quality factor Q =ω0τ.

For Q ¿ 1 the energy dissipated per cycle through one potential minimum is much larger than

the energy gained from the current drive and the junction is strongly overdamped. For this

damping, the particle will immediately stop its motion as soon as the current drive equals

the critical Josephson current, IB = I0. At this point the state of Fig. 2.8(a) is restored and

the particle only oscillates around its zero point. Since the phase is now constant, φ(t) = 0,

the Josephson junction is retrapped in its zero voltage state. In the pendulum analogue,

the dissipation is such strong that as soon as the external drive reaches a critical value, the

pendulum cannot reach the pinnacle of its orbit immediately, so falls back and starts oscillating

around its zero point of motion. In the opposite case Q À 1, the energy dissipated per cycle

is negligible compared to the gain of kinetic energy from the current drive and the junction

is heavily underdamped. In this case, the rolling particle will stop its motion only if the

washboard potential is not tilted anymore. Hence, the retrapping of the Josephson junction

into the zero voltage state occurs at IB = 0, corresponding to a pendulum that stops its rotating

motion when the external drive stops.

In experiments, a likely case will be an intermediately damped junction, corresponding to

Q ≥ 1. In this case the energy dissipated per cycle is smaller than (equal to) the energy gain

from the current drive. In this case, the particle will continue rolling down the tilted washboard

potential, as shown in Fig. 2.8(c), until a minimum tilt is reached. At this particular tilt, the

energy gain from the drive and the energy dissipation cancel out and the particle will stop

its motion. Accordingly, at the corresponding current drive, IR < I0, the junction retraps and

restores its zero voltage state of Fig. 2.8(a). This so-called retrapping current IR can be found

19


Figure 2.9 – (a) Non-hysteretic V (I ) characteristics of a heavily overdamped junction. (b)Hysteretic V (I ) characteristics of a strongly underdamped junction.

from the condition [59]:∫ π

−πdφ

IR

I0= 1

ω20τ

∫ π

−πdφφ(t ). (2.27)

The discussed dependence of the phase dynamics on the amount of dissipation Q can be di-

rectly observed in the experimental V (I ) characteristics of a current biased Josephson junction.

In heavily overdamped junctions, the V (I ) curves are the same when sweeping the current

drive from zero to a current larger than I0 and back to I = 0, since switching and retrapping

occurs at I = I0 (see Fig. 2.9(a)). In contrast, heavily underdamped and intermediately damped

junctions exhibit a hysteresis between forward and backward sweep of the current drive in

the V (I ) characteristics. Figure 2.9(b) shows the extreme case of negligible damping, in which

retrapping occurs at I = 0. For intermediate damping, the retrapping current will be in the

range of 0 < I < I0, as determined from Eq. 2.27.

2.4 Thermal Fluctuations of the Junction Phase φ(t )

In the previous sections, the Josephson effect was derived for an ideal Josephson junction,

isolated from environmental perturbations. However, Josephson junctions in real experiments

are embedded into a measurement circuit that, for instance, contains ohmic leads connecting

the junction. Moreover, real experiments are thermalized at a finite temperature T , so that

these ohmic leads will exhibit thermal resistor noise. This noise induces fluctuations in the

junction phase φ(t ) [31], as it will be presented in the following.

The physical consequences of these thermal fluctuations in φ(t ) can be elucidated by consid-

ering the most simple case of a current-biased Josephson junction, as displayed in Fig. 2.8(b),

with a current bias of I < I0, at a temperature T = 0K. Under these conditions the particle

in the tilted washboard potential will oscillate around the metastable potential minima at

a frequency ω0. If we now switch on the temperature to a finite value and suppose that the

20

2.4. Thermal Fluctuations of the Junction Phase φ(t )

thermal energy E = kB T is comparable to the potential barrier height U , the particle has a

chance to be thermally excited across the barrier, as displayed in Fig. 2.10. The probability of

such an excitation process PTA is proportional to the attempt frequency, which corresponds

to the Josephson frequency ω0, and a Boltzmann factor, which relates the barrier height

U = E J (cos(φ)− I /I0) to the thermal energy kB T :

PT A ∝ω0 exp

(U

kB T

)=

√4e2E J

C Jħ2 exp

(E J (cos(φ)− I /I0)

kB T

). (2.28)

It can be seen from this equation that for a junction of given coupling energy E J and capac-

itance C J the excitation probability increases with increasing current bias I and increasing

temperature T . The dynamic response of φ on such an excitation process depends on the

junction quality factor Q, as shown in Fig. 2.10. If the junction is heavily overdamped, Q ¿ 1,

the excited particle will, in the first place, relax to the adjacent potential minima and according

to φ= 0, the junction remains in its zero voltage state. However, for an increasing tilt, I → I0,

the friction of the particle will be not large enough to compensate for the tilt. Therefore, a

thermal excitation of the particle over the barrier will result in a finite velocity φ and the

junction will switch to the finite voltage state. In the opposite case where the particle does

not experience significant friction in the washboard, Q À 1, even one individual thermal

excitation across the potential barrier is sufficient for the particle to gain a finite velocity φ 6= 0.

For a given thermal excitation level the underdamped junction will, therefore, switch to the

finite voltage state at much smaller bias currents as compared to overdamped junctions [61].

However, in both cases the premature switching out of the zero voltage state reduces the

maximum DC Josephson current to values I < I0 [31, 62].

Figure 2.10 – Coupling potential E (φ) at a current bias of IB /I0 = 0.1. The particle, representingthe junction, oscillates around a potential minimum at a frequency ω0 and can be thermallyexcited across the potential barrier U . If the junction is overdamped Q ¿ 1 the particle willrelax into an adjacent minima (I). For an underdamped junction, the particle will start movingdown the tilted washboard potential (II).

21


The formal context between thermal fluctuation and the Josephson effect was first derived

by Ivanchenko and Zil’berman in the year of 1968 [31]. Assuming thermal white noise on an

ohmic resistor RDC , we can add a noise term δI (t ) to the source current, which modifies the

differential equation for φ in the RCSJ model (see Eq. 2.24) to a stochastic differential equation,

yielding

φ(t )+ 1

τφ(t )+ω2

0 sin(φ(t )) = ω20

I0(I +δI (t )). (2.29)

It is apparent that thermal noise results in a stochastic, additional drive that eventually kicks

the particle out of its zero point position in the potential minimum. For an overdamped

junction where the junction capacitance C J can be neglected, Eq. 2.29 transforms into the so-

called Fokker-Planck equation, which describes the diffusive, i.e. thermally activated, motion

of a particle that experiences a force. Therefore, thermally induced phase fluctuations are

also called phase diffusion processes. The Fokker-Planck equation can be solved analytically

yielding a modified expression for the DC Josephson current [31, 61]:

I (VJ ) = I0ℑ[

I1−ıVJ /β(α)

IıVJ /β(α)

], (2.30)

VJ = 2e

ħd

dt⟨φ⟩ =V − I (V )RW , (2.31)

α = E J

kB T, (2.32)

β = 2e

ħ RDC kB T. (2.33)

Here, Iν(α) denotes the modified Bessel function and RW denotes the DC resistance of the

measurement circuit. A major effect of thermal phase fluctuations on the characteristics of a

Josephson junction is that the DC Josephson effect may be observed at finite voltages VJ across

the tunnel contact as opposed to its observation at zero voltage for an unperturbed junction.

This phenomenon directly results from thermally excited hopping from one metastable state

at φ1 to another state at φ2 corresponding to an average, albeit small, phase velocity φ 6= 0.

Employing the Josephson relation this finite phase velocity translates into a voltage drop

across the tunnel contact 3.

Figure 2.11 displays calculated I (VJ ) characteristics of such a voltage-biased Josephson junc-

tion for different α using Eq. 2.30. The maximum reachable DC Josephson current is smaller

than the critical current and decreases with increasing temperature. Moreover, the peak

shifts to larger voltages with increasing temperatue, which is an intuitive observation, since

the thermal hopping between adjacent potential minima should also be favored at higher

temperatures.

3In accordance to the quantum fluctuation presented in the next section also the occurrence of thermal phasefluctuation does allow for the observation of the DC Josephson effect in voltage-biased experiments.

22

2.5. Quantum Mechanics of a Josephson junction

Figure 2.11 – Calculated DC Josephson currents normalized to I0 for different α values withrespect to the normalized voltage (V0 = I0R) across the tunnel junction. The ohmic resistorwas chosen to be R = 1Ω.

Although Eq. 2.30 was derived for an overdamped junction, the I (V ) characteristics of an

underdamped junction are of similar contour. However, in this case the amplitude of the DC

Josephson current is strongly reduced.

2.5 Quantum Mechanics of a Josephson junction

Within the past sections, the Josephson effect was derived from a many-body representation of

a tunnel contact having two superconducting electrodes. It was also found that the dynamics

of the phase can be described by a classical equation of motion that finds its analogue in

a rotating pendulum. However, the phase φ and the charge q of a Josephson junction are

quantum-mechanical quantities, for which reason these classical interpretations only apply to

special cases, as will be discussed below. It can be shown, that both correspond to quantum-

mechanical operators, φ and q , which obey the commutator relation [63]

[φ, q] = ıe. (2.34)

The lack of commutativity between these two operators signifies that only the charge q or

the phase φ of the junction can be well defined quantum numbers at the same time and,

thus, can be measured with high precision. Hence, the charge and phase exhibit a standard

deviation σq and σφ representing this uncertainty. This purely quantum-mechanical property

has fundamental consequences on the measurement process, as will be discussed in the

following.

To this end, we start out with a Hamiltonian for a simple Josephson junction of capacitance C J

23


employing the operator representation for φ and q [31]

H = q2

2C J−E J cos(φ). (2.35)

Expressing the charge operator by the number operator n, q = n2e, and identifying the

Coulomb charging energy of the tunnel contact EC = 2e2/C J , the Hamiltonian can be rewritten

to

H = 2e2

C Jn2 −E J cos(φ) = EC n2 −E J cos(φ). (2.36)

If we assume an experiment for which E J is much larger than EC , the Coulomb term can be

neglected and the junction phase φ is a good quantum number. In this situation the standard

deviation of the phase vanishes, σφ = 0, and the particle in a washboard potential model is

an appropriate description (cf. Fig. 2.12). Under this condition, the DC Josephson current at

V=0 is carried by coherent Cooper pair tunneling between the superconducting ground states.

In the opposite case, EC À E J , the Josephson coupling term drops out, q is a good quantum

number. We find σq = 0 and σφ =∞. In this situation, the phase is entirely delocalized along

the washboard potential and the Josephson current is carried by the sequential tunneling

of Cooper pairs at finite voltage VJ , which release the excess energy VJ = hν/(2e) into the

environment, i.e. via the emission of photons. At this point it should be noted that the actual

AC Josephson effect also describes the tunneling of Cooper pairs at finite voltage VJ = hν/(2e),

however, for the case of coherent Cooper pair transfer.

In the intermediate situation in which the Coulomb energy is comparable to the Josephson

energy, EC ≈ E J , neither φ nor q are well defined and the standard deviation of both quantities

is non-zero, as shown in Fig. 2.12. In this situation it is not appropriate anymore to represent

the junction phase as a particle but rather by a quantum-mechanical wavefunctionΨ(φ) that

can extend across several minima in the washboard potential. In this situation, tunneling

events through the barrier are possible and result in quantum fluctuations of the junction

phase. These fluctuations have significant consequences. For instance, if the washboard

potential is slightly tilted, these tunneling events yield a tiny but non-zero phase velocity ⟨φ⟩that correspond to a finite voltage φ= (2e/h)V , although after tunneling the junction phase

still remains in a bound state inside an adjacent potential minimum. As a result, the DC

Josephson effect cannot be observed at zero voltage anymore [31]. Moreover, in the regime

EC ≈ E J , both the coherent tunneling of Cooper pairs around zero voltage can be observed as

well as the sequential tunneling to higher voltages [40].

All of these three cases presented induce different experimental consequences:

• E J À EC : The coherent tunneling of Cooper pairs occurs at V = 0 and therefore, the DC

Josephson effect can only be observed in a current-biased Josephson junction.

• E J ¿ EC : Coherent tunneling of Cooper pairs does not occur, but only sequential Cooper

24

2.5. Quantum Mechanics of a Josephson junction

pair tunneling occurs at finite voltages VJ = hν/(2e) and therefore, the Josephson effect

can only be observed in voltage-biased measurements.

• E J ≈ EC : Both coherent and incoherent Cooper pair tunneling occurs, and the DC

Josephson effect is found at very small voltages V 6= 0. Therefore, it is possible to observe

the DC Josephson effect in current-biased measurements, as well as the sequential

tunneling of Cooper pairs at VJ = hν/(2e) in voltage-biased measurements.

A more detailed theory on the quantum mechanics of a Josephson junctions as well as the

possible experimental manifestations can be found in Ch. 6.

Figure 2.12 – Standard deviations of the phase operator σφ and the charge operator σq as afunction of the ratio E J /EC .

25

3 Experimental

The scope of this chapter is to provide the reader with basics of Scanning Tunneling Microscopy

(STM) and to discuss the experimental requirements for performing Josephson STM (JSTM)

experiments. Moreover, this chapter will provide detailed descriptions on the experiment,

preparation of the STM tip and the sample as well as experimental procedures.

3.1 Scanning Tunneling Microscopy

STM is an experimental technique that employs the quantum mechanical tunnel effect (cf.

Sec. 2.2.1) to study the topographic and electronic properties of electrically conducting sur-

faces. Since its first realisation is 1982 by Binnig and Rohrer [64, 65], STM has evolved to

be a major tool in physics and chemistry addressing questions, such as atomic scale mag-

netism and superconductivity, and the electronic properties of individual molecules and their

self-assembly into molecular networks [66, 67, 16, 68, 69, 70, 71, 72]. Moreover, in the recent

decade the experimental parameter space was further extended by the implementation of

time-resolution and as in this work, ultra-low temperatures [32, 33, 34, 35, 36, 73].

The concept of an STM, as displayed in Fig. 3.2(a), involves a sharp electrically conducting

tip that scans across an electrically conducting surface with a vertical separation of a few

Å. During scanning a bias voltage V is applied between tip and sample and the resulting

tunneling current is being recorded 1. In most cases, V corresponds to a generated potential

difference between sample and ground and the measured tunneling current corresponds to

the current, that flows between tip and ground as shown in Fig. 3.2(b).

The central element for topographically mapping a sample surface is the dependence of the

tunneling current on the barrier width (cf. Ch. 2, Sec. 2.2.1). In the year of 1983 Tersoff and

Hamann investigated this dependence for an STM geometry in which the tip is assumed to

have a single atom at its apex [75]. Assuming only s-type orbital wavefunction for the STM

1In the simplest case tip and sample are metals, but it is also possible to use ceramics, as the tunneling electrodesin the context of non-conventional superconductors, or semiconducting samples [65, 74].

27

Chapter 3. Experimental

tip 2, they found that the tunneling matrix element is modified by an additional dependence

on the tip sample distance −→r , Mk,l (−→r ) (cf. Eq. 2.12). The authors could further show, that

Mk,l (−→r ) only depends on the amplitude of the sample wavefunctionΨl (z) at the position of

the tip −→r . In the simplified one dimensional case, the relation between matrix element and

vertical tip sample distance, z ′, reads as

Mk,l (z ′) ∝Ψl (z ′). (3.1)

The resulting dependence of the tunneling current on z ′ can be subsequently found from the

normalization condition for quantum mechanical wavefunctions. The condition

1 =∫ ∞

−∞d z|Ψl (z)|2 (3.2)

implies an exponential decay ofΨl (z) for z →∞ and, therefore, the tunneling current between

tip and sample exponentially depends on their separation z ′,

I (z ′) ∝|Mk,l (z ′)|2 ∝ exp(−2z ′κ). (3.3)

Here, κ denotes the decay constant of the sample’s wavefunction Ψl (z ′) for the decay into

vacuum. Accordingly, if a tip is scanning a perfectly flat surface and reaches an obstacle as

shown in Fig. 3.2(a), an individual atom on top of a surface for instance, the tunneling current

will increase due to the shrinking tip-sample distance.

Scanning the surface topography of real samples is challenging when moving the tip across

the surface at constant height, because it will exhibit obstacles such as atomic steps. Therefore,

it is much more favorable to keep the measured tunneling current constant by adjusting the

distance between tip and sample, which can be done by a so-called feedback loop. When

the tip approaches an obstacle on the surface, the feedback loop detects an increase of the

tunneling current due to the decreasing tip-sample distance. It will immediately lift up the tip

in order to keep the current constant, as it is shown in Fig. 3.2(a) 3. This operation mode, called

constant current mode, facilitates the mapping of the surface topography at high precision and

high velocity and, therefore, represents the standard mode of operation in STM experiments.

It should be mentioned that, in addition to the dependence of the tip sample separation,

the tunneling current also strongly depends on the voltage-dependent local density of states

(LDOS) of the sample ρl (V ) (cf. Eq. 2.16), which can vary for different spots on the sample.

For this reason, a sample topography mapped with STM always includes information of the

local electronic structure and deviates from the real sample topography. In return, the STM

facilitates the investigation of ρl (V ) at the position of the tip, for which reason Scanning

Tunneling Spectroscopy (STS) is the key spectroscopy technique on the atomic scale. ρl can be

2To first order, this is a good approximation, since the s-type orbital wavefunctions extend much further intospace with respect to the center of the atom than p-, d- and f -type wavefunctions [76]

3The control of the tip-sample distance on the atomic scale as well as their lateral motion while scanning thesurface requires piezo actuators, which have sub-picometer accuracy.

28

3.2. Experimental Requirements for JSTM

accessed applying the so-called lock-in technique. This technique adds a small periodic mod-

ulation V (t ) =Vm cos(ωm t ) on top of the applied bias voltage V and measures the resulting

signal amplitude A(ω) at given frequency fm =ωm/(2π). This amplitude is proportional to the

differential conductance, A = dI/dV×Vm and provides direct access to ρl [77].

This presented introduction on the topographic and local spectroscopic capabilities STM

provides the necessary background which is needed in relation to this work, but the interested

reader might refer to standard literature for more details on this technique as given in Refs. [78,

79].

In this work, for all STM experiments the Nanonis SPM controller was used providing all

required properties for the operation of an STM experiment: A voltage bias source, an analog

input for the current signal, high-voltage amplifier for the piezo operation, the feedback-loop

for the control of the tip-sample distance and an internal lock-in amplifier.

Figure 3.1 – (a) Schematics of an STM that scans a sample surface in the x y-direction inconstant current mode. (b) Simplified circuit diagram for standard STM operation using avoltage bias.

3.2 Experimental Requirements for JSTM

Performing JSTM experiments is a challenging task to undertake due to the a variety of high-

end experimental requirements that have to be met. These are a low vibration environment

to enable the operation of an STM, ultra-low temperatures on the order of a few millikelvin

necessary to achieve a sufficient ratio between Josephson coupling energy E J and thermal

energy kB T , E J ≥ kB T , and, last but not least, an overall shielding of the experiment against

radio frequency (RF) radiation, necessary to minimize perturbations to the junction phase φ.

Each of these experimental requirements will be addressed in the following subsections, but a

detailed description of the experimental setup used in this work can be found in Refs. [80, 81,

35].

29


3.2.1 Mechanical Stability

As is was introduced in the previous section, the tunneling current between the STM tip

and the sample exponentially depends on their distance, for which reason it is not only

sensitive to the atomic scale topography of the sample, but also to mechanical disturbances

imposed by the measurement environment. Such disturbances originate from vibrations of

the laboratory building structure, from impact noise of scientists walking around or from a

pumping environment necessary for operating a cryogenic instrument, for instance. These

mechanical motions are transferred along the experimental setup into the tunnel junction

of the STM experiment and can result in oscillations of the tip-sample distance, so that STM

operation is impossible. Therefore, an STM setup demands both a sturdy construction of the

experimental structure as well as an isolation of this structure from external mechanical noise

sources.

Figure 3.2 – (a) Laboratory hall and damping stages: (I) Two story high noise-canceling room(II) separate noise-canceling pumping room (III) Actively-damped stage (red) and passively-damped stage (green), on which the cryostat (green) is mounted. The pre-damping of thetubing is shown in blue color. (IV) Ultra-high vacuum (UHV) preparation chamber attachedto the bottom of the cryostat. (b) Custom designed STM head [80]: (I) Coarse-motion [82] (II)Piezo-stack for x y z-control, containing the tip. (III) Sample-thread with bias voltage contact(IV) Transferable sample. Both figures are adapted from Ref. [35].

The STM used for this work combines both active and passive damping elements in order to

minimize disturbances to the tip-sample distance [83, 84, 81]. Fig. 3.2(a) displays the two story

30


high laboratory hall of the millikelvin STM experiment. The cryostat containing the STM head

sits on top of an active piezo-driven damping stage and a passive, air spring damping stage,

which isolates the experimental setup from the lab floor. Moreover, operating a continuous

flow dilution refrigerator system, as it will be discussed in the next subsection, requires

circulator pumps and tubings, which serve as perfect mechanical noise sources and noise

conductors. For this reason, these pumps are placed in an additional noise-encapsulating

room and the tubings connecting the cryostat to the pumps are pre-damped on an active

damping stage and contain flexible elements hindering vibration transfer.

The STM measurement head is custom made in a specialized design so as to minimize the

response of the tip sample distance to mechanical vibrations [80]. Its cross section displayed

in Fig. 3.2(b) presents all elements that are necessary for the STM operation: A coarse-motion

to approach the sample from millimeter distance to less than a micrometer. A piezo-stack

that facilitates x y z-motion with sub-picometer accuracy over a range of few hundreds of

nanometers. Moreover, the custom design allows for the in-situ exchange of STM tip and

sample, a property of upmost importance for this work as it will be demonstrated later.

In this configuration the stability of the STM experiment, defined by the standard deviation of

the average tip-sample distance, σz ′ , was measured to σz ′ = 4.2±0.5pm at a temperature of

15 mK [35].

3.2.2 Refrigeration to Millikelvin Temperatures

Experimentally investigating phenomena whose energy scale is much smaller than the ther-

mal energy at room temperature, kB T ≈ 26meV, requires the application of refrigeration

techniques. Superconductivity in mercury, for instance, was first discovered by Kammerlingh-

Onnes in 1911 when he succeeded in liquefying helium [85]. Attaching a reservoir of liquid

helium to the experiment enables refrigeration down to temperatures of about T =4 K which

corresponds to a thermal energy of ≈ 0.1meV 4. For this reason liquid helium served as the

preferred refrigeration medium over the last century. Studying the Josephson effect in tunnel

junctions, however, requires even lower temperatures since the coupling energy of Josephson

tunnel junctions in STM experiments is very small, E J /kB ¿ 1K 5. Therefore, cooling with

liquid Helium is not sufficient anymore and alternative refrigeration techniques have to be

employed such as the dilution refrigerator [86, 87].

The dilution refrigerator (DR) exploits the latent heat of mixing the two helium isotopes 4He

and 3He and allows for the refrigeration of experimental setups down to temperatures of a

few millikelvin [88]. The operation procedure will be briefly described in the following, but

a more comprehensive discourse can be found in standard literature, e.g Ref. [88]. When a

mixture of 4He and 3He atoms is cooled below a temperature of T = 0.867K, it will separate

into two different phases that float on top of each other. The upper concentrated phase only

4Even temperatures of about T =1 K can be reached experimentally when exploiting helium’s latent heat.5For junctions using elemental superconductors having an order parameter ∆≈ 1meV

31


consists of 3He atoms, whereas the heavier diluted phase on the bottom mostly contains 4He

atoms plus a fraction of 3He atoms 6. This finite content of 3He in the dilute phase displays an

energetic minimum of the system at given temperature T and is of fundamental importance

for the refrigeration process 7: Removing 3He atoms from the diluted phase forces other 3He

atoms in the concentrated phase to cross the phase boundary into the diluted phase. The

transfer process increases the entropy of the system, thus requires energy. This energy is taken

from the environment in the form of thermal energy and in this way the environment, e.g. an

experimental setup, is being refrigerated [88].

The described mixing process can be operated in a continuous mode such, that 3He atoms

are extracted out of the diluted phase and recycled into the concentrated phase by means of

circulator pumps, as schematically shown in Fig. 3.3(a) 8. In this way, it is possible to generate

sufficient cooling power for continuous refrigeration down to temperatures below 10 mK.

The cryostat used for this experiments from Janis Research Company has a cooling power

of Q ≈ 400µW at a temperature of T ≤ 100mK [81, 35, 89]. This cooling power is relatively

low compared to helium bath cryostats, for which reason the DR cryostat has to match

specific requirements in order to reach millikelvin temperatures. Figure 3.2(b) displays a

cross-section of the DR cryostat, which is constructed in analogy to onion skins: An outer

liquid nitrogen dewar shields the liquid helium dewar on the inside from thermal radiation at

room temperature. In the same fashion, this dewar shields the DR stage containing several

temperature zones ranging from ≈ 1.5K down to the mixing chamber at T = 15mK, where

the refrigeration process takes place. All temperature stages are used in order to liquefy

and pre-cool the 3He gas. It should be noted, that the pre-cooling process is of essential

importance in order to reach temperatures below 50 mK, and this process bundles most of

the required knowledge for building a state of the art DR stage. The STM (cf. Fig 3.2(b)), is

attached to the mixing chamber via an thermally conductive scaffold manufactured from

silver and thus reaches the same base temperature of 15 mK [35]. The employed DR system

allows continuous refrigeration of the STM head to temperatures between 15 mK and 800 mK.

However, all experiments in this work were performed either at 15 mK or 800 mK, respectively.

3.2.3 Electronic Temperature and Energy Resolution

Reaching very low experimental temperatures on the order of millikelvin is feasible because

nowadays DR cryostats that reach temperatures as low as 10 mK are commercially available.

However, the physically measured temperature at the mixing chamber in first approximation

only represents the temperature of the crystal lattice. The temperature of the electrons moving

6Due to the larger mass, 4He atoms have a smaller zero point motion as compared to 3He atoms and, thus, thediluted phase is of higher density than the concentrated phase and floats at the bottom.

7This minimum displays the point of equivalent attractive and repulsive interaction potentials of 3He atoms inthe diluted phase. The interactions are of both thermodynamic and quantum-mechanical origin, which resultfrom the fermionic nature of the 3He isotopes.

8The still chamber is operated at a temperature of T ≈ 800mK . At this temperature the gas pressure of 3He ismuch larger compared to the gas pressure of 4He. For this reason, only 3He is cycled effectively.

32


Figure 3.3 – (a) Schematics of the dilution refrigerator: A rotary pump (I) cycles 3He atomsfrom the diluted phase in the so-called st i l l-chamber (II) to the concentrated phase in themixing chamber (III). (b) Cross-section of the Janis DR cryostat: (I) liquid nitrogen dewar (II)liquid Helium dewar (III) DR stage (IV) STM head attached to the mixing chamber (V) Shutterfeedthroughs to the ultra-high vacuum (UHV) preparation chamber. (c) RF filtering concept:All wires into (out of) the cryostat are RF filtered at the UHV feedthrough. Secondly, the wiresare thoroughly thermalized (black knots) at the several temperature stages of the DR, at (iv)1.5 K, (iii) 800 mK, (ii) 100 mK and (i) 15 mK. The black cross marks the position of the STMhead.

through the lattice might deviate since the electron-phonon coupling, which mediates the

cooling energy transfer, is very weak in this temperature range. Moreover, at T = 15mK thermal

energy is so small, 15mK ≈ 1.3µeV/kB ≈ 300MHz/(kBħ) that any thermal radiation from the

environment as well as RF noise from the atmosphere represent a serious energy load that

can heat up the electronic bath substantially. Hence, without further experimental means the

electron bath in the junction leads remains ’hot’ on the order of hundreds of millikelvin such

that it will be impossible to observe phenomena whose energy scales are smaller, even when

using a DR cryostat. Therefore, it is necessary to achieve a thorough thermalization of the

wiring, a proper shielding from thermal radiation and filtering from RF noise to also obtain

the lowest electronic temperatures.

In the experiment used for this study, thermalization of all wires is achieved by thermally

anchoring every single wire at every temperature stage of the DR, as shown in Fig. 3.3(c).

Moreover, at each of these thermalization points the wires are wrapped around bobbins in

33


order to increase the heat energy transfer effectiveness so that the small electron-phonon

coupling can be overcome. Thermal radiation is reduced by shielding the mixing chamber as

well as the STM head from temperatures larger than T = 100mK or T = 800mK, respectively.

Furthermore, the STM head itself is a closed metal body of large mass thermalized at T =15mK, which provides additional shielding of sample and tip to heat radiation of 800 mK

(cf. Fig. 3.2(b)). All wires leading to the STM head are continuously thermalized as well as

protected by radiation shields manufactured out of silver [81]. Shielding from RF radiation

is achieved by three different components (cf. Fig.3.3(c)): First of all, the cryostat itself is a

fully closed metal body acting as a perfect Faraday cage that cancels out any electromagnetic

radiation. The only openings are the wire feedthroughs, for which reason all of them, signal-

and control-lines, are low-pass filtered for RF radiation above 10 kHz [81, 35]. Moreover, the

wires in the cryostat themselves, reaching down to the mixing chamber stage, are distributed

RF filters having an attenuation of -120 dB/m for frequencies above 100 MHz [90, 81].

Another aspect of relevance for STM experiments at millikelvin temperatures that has not

been addressed so far is the limitation of the electronic resolution due to thermal voltage

noise u on the STM junction capacitance C J . This voltage noise results from thermal charge

fluctuations Q, Q =C J ×u, and depends on the temperature as [91, 92]

u =√

kB T

C J. (3.4)

The capacitance value of an STM tunnel contact can be easily estimated by electrostatic means

when applying a simple 3D cone model for the STM tip in front of a planar surface, which

mimics the sample. For the 250µm thick wire used in this study such a calculation yields

a capacitance of C J ≈ 5fF. According to Eq. 3.4, we can calculate the thermal voltage noise

on the STM junction capacitance at 15 mK to u = 6.4µeV. This finding signifies that even in

the unlikely case that the electrons in the junction leads reach the exact lattice temperature,

the small capacitance of the STM junction limits the energy resolution of STM experiments

to ∆E ≥ 6.4µeV 9. For the sake of clarity, the term energy resolution denotes the minimum

energetic difference ∆E between two energy levels that can still be resolved by means of

spectroscopy, e.g. STS (cf. Sec. 3.1). It should be noted, that the calculated value only represents

an estimate for standard STM conditions. Constructing STM junctions of larger capacitance

will significantly decrease the thermal voltage noise amplitude, which is of interest in the

context of the AC Josephson spectrometer at the atomic scale (cf. Ch. 5).

The energy resolution of the STM used for this work was investigated via thermal excitations

of quasiparticles in superconducting aluminium tips. It was determined to ∆E = 11.4±0.3µeV

[35]. This value contains both the capacitive noise as well as the electronic temperature but,

due to a lack of precise knowledge of C J , the effective electronic temperature itself remains

unknown.

9The voltage noise on external capacitances, such as capacitances used for filtering, is smaller in our case,u′ ≤ 1µeV

34

3.3. Sample and Tip Preparation

3.3 Sample and Tip Preparation

Investigating sample properties with an STM, such as the surface topography of a metal or

individual atoms and molecules, also requires the capability to prepare these atomic scale

sample systems. Moreover, both the preparation as well as the STM experiment have to

be performed in an environment that avoids any contamination of the surface with other

adsorbates, such as molecules from the surrounding atmosphere. Therefore, the preparation

chamber of the system as well as the cryostat used for this work, as shown in Fig. 3.2(a), are

operated at UHV conditions at a base pressure of p ≤ 5×10−10 mbar. After the sample has been

prepared, our experimental setup enables a sample transfer into the STM head (cf. Fig. 3.2(b))

without breaking the UHV.

Experiments involving the Josephson effect require two electrodes, whose superconducting

properties are well described in the framework of BCS theory (cf. Sec. 2.1). All experiments in

this work were performed using vanadium as the tip and the sample material. Vanadium is an

elemental type-II BCS superconductor with a critical temperature of T = 5.4K [93]. Details of

the preparation procedures will be part of the following subsections.

3.3.1 Preparation of Vanadium (001) Single Crystal

For the experimental study of this work we use a vanadium(001) single crystal of purity 6N ,

which was purchased from SPL [94]. Atomically flat as well as impurity free surfaces can be

obtained by means of ion sputtering and thermal annealing. To this end, the sample surface is

sputtered with argon ions, which have a kinetic energy of E = 1keV, at an argon pressure of

p = 1.6×10−6 mbar and for a maximum time of 1 hour. Subsequently, the sample is annealed

to a temperature of T ≤ 900C. These two steps, sputtering and annealing, are continuously

cycled as soon as the chamber pressure during the annealing step reaches values smaller

than 2×10−9 mbar. For more details on these standard surface preparation procedures, the

author encourages the reader to refer to standard literature, e.g. Ref. [95]. Figure 3.4(a) displays

the freshly prepared vanadium surface on a macroscopic scale. It features large atomic layer

terraces and moreover, the topography itself features a strip-like structure. The dark spots are

surface defects which might originate from an empty vanadium lattice site in the top layer.

Figure 3.4(b) displays the close-up on the atomic scale, in which the single vanadium atoms as

well as a periodic superstructure of the atoms is apparent. The superstructure corresponds to

the oxygen-assisted (5×1) surface reconstruction of the vanadium (001) surfaces, as reported

previously [96, 97]. Such a clean sample surface allows us to create a high-quality Josephson

junction between the STM tip and sample with a vacuum tunnel barrier whose width can

be precisely tuned by the tip-sample distance. In particular, the clean surface and the point

like tunnel junction of the STM excludes possible disorders, a non-uniform tunnel barrier

thickness or electric shorts through the tunnel barrier for instance, as they can occur in planar

junction geometries.

35


Figure 3.4 – (a) Large scale 100×100nm2 topography of the vanadium (001) surface measuredat V = 1V and I = 50pA. (b) Atomic scale surface reconstruction of the same vanadium sample(4×4nm2) measured at V = 2mV and I = 5nA

3.3.2 Preparation of Vanadium Tips

The STM tip is manufactured from a 250µm thick, polycrystalline vanadium wire of 3N purity

that was purchased from Alfa-Aesar. The tip is prepared by cutting the vanadium wire under

tension. After the tip is mounted into the tip holder it is immediately transferred into the

UHV environment and the STM head. To remove the native oxide, we perform field emission

between the vanadium tip and the vanadium sample for t ≤ 20min at a field emission current

of IF E ≈ 20µA when a voltage VF E ≈ 50V is applied. Moreover, we apply voltage pulses of

amplitude −8 ≤ V ≤ 8V between the tip and the sample until both of them exhibit a well-

developed BCS gap, as displayed in Fig. 3.5.

3.4 Measurement Circuit

Josephson junctions of which the capacitive charging energy EC is comparable to the Joseph-

son coupling energy E J , can be investigated by means of both voltage-biased and current-

biased experiments, as discussed in Sec. 2.5. In voltage-biased measurements, a voltage bias

VB is applied and the resulting tunneling current I (V ) is measured. The circuit diagram of

such measurements is shown in Fig. 3.6(a), where the voltage source is the Nanonis SPM

controller. The current is measured with a trans-impedance amplifier [98], which converts the

current signals into voltage signals. These voltage signals are recorded by the Nanonis SPM

controller analog input. The available minimum voltage step of ∆V ≥ 150µV results from the

limited bit resolution of the analog-to-digital converter of the Nanonis voltage source, so that

an additional voltage divider 1/1000 is used. In this work, all voltage-biased experiments on

Josephson junctions are performed at a bias voltage step width of ∆V = 1µV across a range of

36

3.4. Measurement Circuit

Figure 3.5 – d I /dV spectrum of a vanadium-vacuum-vanadium junction measured at T =800mK normalized to ρN (V = 2mV, I = 500pA, fm = 720Hz, Vm = 20µV).

typically −500µV ≤V ≤ 500µV, unless stated otherwise.

To gain knowledge on the superconducting order parameter ∆ of the tip and the sample, we

also investigate the quasiparticle excitation spectrum of the tunnel junction. To this end, we

record d I /dV spectra using the internal lock-in amplifier of the Nanonis controller. All of

these measurements are performed at a voltage bias step width of ∆V = 8µV across a range of

−2mV ≤V ≤ 2mV and using the lock-in parameter fm = 720Hz and Vm = 20µV.

Figure 3.6 – (a) Voltage bias setup: The applied voltage, VB , corresponds to a generatedpotential difference between sample and ground. A voltage divider, VD, refines the bias stepwidth and the bias signal is RF filtered (RF). The tunneling current I is measured using atrans-impedance amplifier. (b) Current bias setup: The SMU generates a current bias IB ,which is RF-filtered before it is fed into the cryostat. The voltage drop between current sourceand ground, V , is also measured using the SMU.

In current-biased experiments a current IB is forced through the tunnel junction and a voltage

drop, V (IB ) is being measured. Since the Nanonis SPM controller does not feature a bias

37


Table 3.1 – Circuit resistance of voltage biased measurements for different amplification rangesof the current amplifier

Current range ≤10 nA ≤100 nA ≤ 1µARDC ,1(kΩ) 14 4.8 3.9

current output, an external source measure unit (SMU) is employed for both biasing the

current and measuring the voltage drop across the tunnel junction in a two port measurement.

To this end, the usual circuit to the Nanonis is replaced by a circuit containing the SMU, as

shown in Fig. 3.6(b). Under these conditions, the feedback-loop is no longer applicable, but

the tip-sample distance can still be controlled manually with the z-piezo voltage. Hence, the

conductance GN of the tunnel contact can still be adjusted to the desired value by, GN (z ′) =IB /V ∝ exp(z ′κ)/V . The SMU used for this study [99] features a minimum current step width

of ∆I = 10fA and resolves voltages of ∆V ≥ 100nV. If not stated otherwise, all measurement

were performed on a bias current range of logarithmic point density, so that around I = 0A,

the minimum current step width corresponds to the source limit of 10 fA.

Whether the voltage is applied as a bias VB or measured as a signal V , in both cases it does not

correspond to the voltage drop across the tunnel junction VJ , but to the voltage drop across

the entire measurement circuit. In most cases such a circuit contains several resistive elements

RDC , e.g. wires, filters or the voltage divider. The voltage drop across the tunnel junction VJ is

obtained by subtracting these voltage drops from the source voltage or the measured voltage,

respectively, which reads as

VJ =V(B) − I (V(B))×RDC ,i . (3.5)

From measurements we find a circuit resistance of RDC ,2 = 3750Ω for the current-biased

measurements. The circuit resistance of the voltage-biased measurements, RDC ,1, depends on

the gain of the current amplifier as summarized in Tab. 3.1.

3.5 Experimental Procedure

The past sections presented the experimental requirements for performing JSTM experiments

at ultra low temperatures. The scope of this section is to briefly summarize the experimental

procedure from the sample preparation to the actual measurement.

The first step of our experiment is the transfer of the STM tip and the sample into the cryostat

after they have been prepared as described in Sec. 3.3. Since the tip and the sample are

thermalized at temperatures between 77 and 273 K, their transfer into the cryostat introduces

a serious heat load that will warm up the cryostat to temperatures of around 77 K. After

the cryostat is refrigerated again to a temperature of about 800 mK – which usually takes

approximately one night – we approach the tip to the sample using the coarse motor. As

soon as the tip is in contact with the sample, we remove the native vanadium oxide from

38

3.5. Experimental Procedure

the tip apex by performing field emission (cf. Sec. 3.3). The field emission process leaves a

lot of debris on the entire accessible surface of 1×1µm2 area 10, for which reason after field

emission we retract the tip and rotate the sample, so that the tip will approach a different and

clean sample spot. After refrigeration to 800 mK we again approach the sample. As soon as

the contact is reached, we apply voltage pulses until both the tip and the sample exhibit the

well developed superconducting BCS gap, as shown in Fig.3.4. At this point, we start with the

refrigeration process by increasing the amount of 3He-4He-mixture in the DR cryostat. It takes

typically about one night until the mixing chamber and the STM head are thermalized at the

base temperature of 15 mK. As soon as we have found an atomically clean surface, e.g. shown

in Fig. 3.4, we start with the current- and voltage-biased experiment as explained in Sec. 3.4.

From these experiments we obtain the experimental data, which we will discuss in the next

three chapters.

10The edge length of this square is limited by the maximum piezo displacement.

39

4 The Critical Josephson Current in theDynamical Coulomb Blockade Regime

The content of this chapter is partly taken from the manuscript entitled “Critical Josephson

Current in the Dynamical Coulomb Blockade Regime“ by the authors Jäck, B., Eltschka, M.,

Assig, M., Etzkorn, M., Ast, C.R. and Klaus Kern, which we recently submitted for publication.

4.1 Introduction

The critical Josephson current I0 is a universal parameter that characterizes the coupling

configuration between two superconducting ground states in the tunnel junction electrodes.

In the tunnel regime, i.e. the normal state conductance GN of the tunnel contact is much

smaller than the quantum of conductance G0, the critical Josephson current directly depends

on GN and the superconducting order parameters∆1 and∆2, as it was derived by Ambegaokar

and Baratoff (cf. Eq. 2.21)[8]. Thus, the Josephson critical current provides direct access to the

superconductor ground state and for this reason, it serves as an ideal probe for superconductor

physics, such as the pairing symmetry of unconventional superconductors [100]. The desirable

combination of the Josephson effect with Scanning Tunneling Microscopy (STM), also referred

to as JSTM, transfers this capability to the atomic scale [14]. Additionally, this combination

enables the investigations of local variations of the superconducting order parameter as a

probe for a variety of phenomena. These are, for instance, the response of the superconducting

ground state to the local dopant structure in unconventional superconductors [101] or the

interaction of the superconducting ground state with a single magnetic impurity [16]. In

particular the competition between superconductivity, which favors pairing of electrons with

opposite spin directions, and magnetism, which favors pairing of electrons with same spin

direction, on the local scale has attracted theoretical [102, 18, 17] as well as experimental

[16, 20] efforts for decades, also in the context of unconventional superconductors [103].

So far, experiments exploiting the JSTM concept [26, 27, 28, 29, 30] have measured at compa-

rably high temperatures, where the thermal energy exceeds the Josephson coupling energy,

so that the critical current cannot be accessed unambiguously [31, 104]. Directly accessing

the critical current in an experiment requires special design and tuning of the properties of

41

Chapter 4. The Critical Josephson Current in the Dynamical Coulomb Blockade Regime

the Josephson junction beforehand or it demands detailed knowledge about the junction

parameters as well as tuning of the involved energy scales for precise modeling. While the

former approach has been successfully applied in planar tunnel junctions [39, 61, 62], other

experiments, such as STM, have much less design flexibility. In these cases, the latter approach

is more appropriate, but not every regime in which a Josephson junction is operated can

be modeled equally well for extracting the critical current. For instance, the Ivanchenko-

Zil’berman model, discussed in Sec. 2.4, is only valid for systems of large capacitance and a

frequency-independent circuit impedance Z (ν) = Z (0). This clearly does not apply to STM

experiments that involve a small tunnel junction capacity and a junction environment that

contains transmission lines with strong frequency dependence Z (ν) [31].

Choosing the appropriate theoretical model in fact requires a careful consideration of all

involved phenomena that effect the properties of the Josephson junction. These are the

Josephson coupling energy E J =ħI0/(2e), the Coulomb charging energy of the tunnel contact

EC = 2e2/C J , where C J is the junction capacitance, as well as the thermal energy ET = kB T ,

where T is the temperature. Fig. 4.1 compares these energy scales as a function of the normal-

ized tunneling conductance GN /G0 for our STM experiment. We find that in the tunneling

regime, GN ¿G0, the energy scales order in the following way: ET ≤ E J ¿ EC . This reduces

thermal fluctuations of the junction phase as introduced in Sec. 2.4 and, in particular, this

means that the condition ET ≤ E J for JSTM to work best is fulfilled for most of the tunneling

conductance range [31, 14].

Figure 4.1 – Comparison of all relevant energy scales as a function of the normalized tunnelingconductance for a Josephson junction in an STM: E J denotes the Josephson coupling energy,EC the capacitive charging energy and kB T the thermal energy.

In addition, we find E J ¿ EC and can conclude that the tunneling current in our STM exper-

iment is carried by the sequential tunneling of Cooper pairs. In this regime, E J ¿ EC , also

referred to as the dynamical Coulomb blockade (DCB) regime, the DC Josephson effect can not

be observed at V = 0 anymore (cf. Sec. 2.5) [31, 37, 38]. Instead, Cooper pairs tunnel at finite

42

4.2. P (E)-Theory

voltages ∆V 6= 0, which results in a so-called supercurrent peak, as shown in Fig.4.2 [62]. Since

Cooper pairs consist of two electrons of opposite momentum and opposite spin orientation,

they carry no kinetic energy as well as no net spin. Accordingly, the Cooper pair tunneling at

finite voltages represents an inelastic process, in which the energy quanta hν, proportional

to the junction bias voltage VJ = hν/(2e), are released into the environment via the emission

of a photon of energy E = hν (compare Fig. 4.3). The emitted photon spectrum has recently

been studied in more detail [105], also in the context of non-linear quantum electrodynamics

[106]. Moreover, a junction environment that exhibits spectral resonances, e.g. standing

electromagnetic modes of energy E = hν, can interact with the Josephson junction and induce

inelastic Cooper pair tunneling at an applied voltage bias 2eVJ = hν 1 [107, 39]. Such a process

manifests itself as additional spectral resonances in the measured I (VJ ) characteristics and

these resonances are also present in our experiment, as displayed in Fig. 4.2.

Figure 4.2 – Typical I (VJ ) tunneling characteristics measured at GN /G0 = 0.27 and T = 15mK.

The inelastic Cooper pair tunneling characteristics of voltage-biased Josephson junctions

can be analyzed in the framework of P (E)-theory [37, 38]. This theory models the spectral

energy exchange probability P (E) between a tunneling Cooper pair and an – quite general

– environment, from which the inelastic tunneling current I (VJ ) can directly be calculated.

The following section will present the derivation of P (E )-theory that is necessary to model the

Cooper pair tunneling characteristics of a voltage-biased Josephson junction in an STM at

millikelvin temperatures.

4.2 P (E)-Theory

The theoretical basis for calculating the tunneling characteristics of microscopic tunnel junc-

tions was developed along with the technological advances in manufacturing such micro-

scopic devices on the sub-micrometer length scale in the 1980s. In analogy to the theory

1The factor of two here considers the double electron charge of the Cooper pair.

43


Figure 4.3 – Illustration of inelastic Cooper pair tunneling in the dynamical Coulomb blockaderegime.

of the tunneling effect in macroscopic junctions (cf. Sec. 2.2), also the tunneling theory for

microscopic junctions was derived without limitations to superconducting tunneling. In fact,

the central element of this theory is the quantum nature of the junction phaseφ and the charge

q (cf. Sec. 2.5), which are of great relevance for microscopic devices in which quantum effects

come to light. The second element is a formalism that describes the dissipative interaction

of a quantum system with an arbitrary environment, which was first formulated by Caldeira

and Leggett in 1983 [108]. They could show that a dissipative environment corresponds to

a set of N harmonic oscillators represented by a set of N LC -circuits of capacitance Cn and

inductance Ln . The Hamiltonian of this type of environment then reads as

H =N∑

n=1

[q2

n

2Cn+

(ħe

)2 1

2Ln

(φ

2− e

ħV t −φn

)2], (4.1)

whereas in the limit N → ∞ the environment can be rewritten in terms of a frequency-

dependent impedance

Z (2πν) =[∫ ∞

0d t exp(−ıνt )

N∑n=1

cos(νn t )

Ln

]−1

. (4.2)

As we will see later, the formulation of the environment as a complex-valued impedance

Z (ν) will be of great advantage in order to model experimental data. It should be noted, that

the achievement of Caldeira and Leggett reaches far beyond the research area of Josephson

physics discussed in this work, since energy dissipation from quantized energy levels into a

continuous environment is found in any quantum mechanical system and, moreover, is still

part of ongoing research.

In order to calculate the tunneling current of a microscopic Josephson junction, the junction

Hamiltonian (cf. Sec. 2.5) has first to be extended by the environment [107, 109]

H = q2

2C J−E J cos(φ(t ))+

N∑n=1

[q2

n

2Cn+

(ħe

)2 1

2Ln

(φ

2− e

ħV t −φn

)2]. (4.3)

As has been already introduced before, in the dynamical Coulomb blockade regime the

44

4.2. P (E)-Theory

Coulomb charging energy of the junction EC exceeds the Josephson coupling energy, E J ,

which represents the Cooper pair transfer across the tunnel junction. Therefore, the tunneling

rate of Cooper pairs can be evaluated by means of time-dependent perturbation theory,

treating E J as a perturbation to EC2.

Applying Fermi’s golden rule the voltage-dependent tunneling rate Γ(VJ ) can be evaluated to

Γ(VJ ) =E 2

J

4ħ2

∫ ∞

−∞d t exp(ıνt )⟨exp(ıφ(t ))exp(ıφ(0))⟩. (4.4)

The entire interaction between junction and environment, facilitating the inelastic tunneling

process, is contained in the so-called phase-phase correlation function J (t ) = ⟨exp(ıφ(t ))exp(ıφ(0))⟩.This function represents the average change of the phase coherence via the interaction with

the environment and accordingly the transfer of Cooper pairs (cf. Sec. 2.2.5). The correlation

function can be analytically evaluated for an arbitrary environment and reads as [107, 109]

J (t ) = 2∫ ∞

−∞dν

ν

ℜ[Zt (ν)]

RQ

exp(−ıνt )−1

1−exp(−hν/(kB T )). (4.5)

J(t ) is dominated by the frequency-dependent environmental impedance Zt (ν) normalized

to the resistance quantum of Cooper pairs RQ = h/(2e)2. Here, Zt (ν) represents the total

environmental impedance that contains both the environment and the junction capacitance

C J , which acts as a capacitive shunt:

Zt (ν) = 1

ı2πνC J +Z−1(ν). (4.6)

Moreover, J(t) explicitly contains the temperature of the environment, which causes fluc-

tuations to the junction phase φ(t) (cf. Sec. 2.4). By separating the interaction with the

environment and the Josephson coupling of the junction, it is now possible to rewrite the

expression for the tunneling rate, Eq. 4.4, yielding

Γ(VJ ) =πE 2

J

2ħ P (2eVJ ), (4.7)

P (E) = 1

2πħ∫ ∞

−∞d t exp

[J (t )+ ı

ħEt]

. (4.8)

This P (E)-function, serving as the theory’s name giver, represents the spectral probability of

a Cooper pair to exchange an energy portion E = 2eVJ with the environment ZT (ν) when a

voltage bias 2eVJ is applied. Since the probability distribution for the energy exchange P (E ) is

directly proportional to the tunneling rate, it follows that such an inelastic energy exchange

between Cooper pair and environment directly results in a tunneling current signal. In other

words, only in the case where the environment provides a possibility for the Cooper pair to

release its excess energy hν= 2eVJ (or the opposite case the environment provides the energy

2For a detailed description of the theoretical procedure the reader might refer to Refs. [107, 109]

45


portion 2eVJ = hν), Cooper pairs can tunnel and, correspondingly, a tunneling current can be

measured at the applied voltage 2eVJ .

It should be noted that the total energy exchange probability over the entire spectral range is

unity, for which reason it requires normalization of the calculated P (E)-function, reading

1 =∫ ∞

−∞P (E)dE . (4.9)

Finally, the total tunneling current through a microscopic Josephson junction can be calculated

as the difference between the probabilities of releasing energy into the environment, P (2eVJ ),

and of absorbing energy from the environment, P (2eVJ ), respectively [37, 110, 111]:

I (VJ ) =πeE 2

J

ħ [P (2eV )−P (−2eV )]. (4.10)

The above presented derivation of the P (E)-theory reveals several essential properties of this

approach from which an analysis of experimental data will greatly benefit:

• In order to calculate the energy exchange probability distribution P (E) of a specific

junction environment, it only requires a complex expression of the environmental

impedance Z (ν) as the sole input parameter.

• All interaction of the junction with the environment is contained in the P (E ) distribution,

which is independent of the tunneling conductance GN . Hence, for all values of GN the

P (E) distributions will be the same.

• The Josephson effect itself enters the theory only as a scaling factor E 2J of the previously

calculated P (E) distribution. Hence, the Josephson coupling energy E J = (ħ/2e)I0 can

be determined independently with high accuracy.

For the purpose of JSTM the Josephson critical current I0 has to be determined from exper-

imentally determined Cooper pair tunneling spectra I (VJ ). On the one hand, E J = (ħ/2e)I0

is an independent scaling parameter favoring this goal. On the other hand, it still requires

fitting of a calculated P (E) distribution to the experimental data which is quite a difficult

task to undertake. Only for very few cases of simple environmental impedance functions and

in the zero temperature limit, can the P (E) distribution be calculated analytically [37, 109].

However, this does not represent the most common case found in experiment, in which at

least temperature will be non zero. For this reason the P (E)-function for the interaction with

the environmental impedance Zt (ν) has to be calculated by means of numerical computation.

In addition, other phenomena, such as thermal voltage noise u on the junction capacity, can

serve as an energy source or drain for the inelastic tunneling of Cooper pair. In order to cor-

rectly model experimental Cooper pair tunneling characteristics it is necessary to include all

46

4.2. P (E)-Theory

possible inelastic interactions with the environment. Therefore, the presented description of

the P (E )-theory requires further extensions, which will be be part of the following subsections.

4.2.1 Numerical Computation of the PZ (E ) Distribution for an Arbitrary ImpedanceZt (ν)

The numerical calculation of the PZ (E ) distribution that models the interaction of a Josephson

junction with an arbitrary environmental impedance Zt (ν) could, in principle, be achieved by

first computing Eq. 4.5 and 4.8 3. Still, the numerical calculation of these functions is difficult,

for which reason Ingold and Grabert derived an integral equation [112]

PZ (E) = I (E)+∫ E0

−E0

dνK (E ,ν)P (E −hν), (4.11)

which can be solved iteratively and enables the numerical calculation of the PZ (E ) distribution

[112].

I (E) = 1

πh

D

D2 +E 2/h2 , (4.12)

D = πkB T

h

ℜ[Zt (0)]

RQ. (4.13)

The inhomogeneity of Lorentzian shape, I (E), represents a probability distribution that de-

termines the minimum width of PZ (E), as a result of thermal noise on an ohmic resistor

RDC =ℜ [ZT (0)] in the junction environment. The integral kernel K (E ,2πν) contains all the

interactions with the frequency-dependent environment:

K (E ,ν) = E/h

D2 + (E/h)2 k(ν)+ D

D2 + (E/h)2 κ(ν), (4.14)

k(ν) = 1

1−exp[hν/(kB T )]

ℜ[Zt (ν)]

RQ− kB T

hν

ℜ[Zt (0)]

RQ, (4.15)

κ(ν) = 1

1−exp[hν/(kB T )]

ℑ[Zt (ν)]

RQ− 2kB T

h

∞∑n=1

ωn

ω2n +ν2

n

Zt (−ıωn)

RQ. (4.16)

Here ωn = 2πnkB T denotes the Matsubara frequencies. This set of equation enables the

self-consistent calculation of a PZ (E ) distribution, where the inhomogeneity I (E ) serves as an

initial test function Pm=0(E ). Employing the environmental impedance, as it will be introduced

in Sec. 4.3, the integral equation, Eq. 4.11, can be solved iteratively until self-consistency is

reached with a relative deviation of 1×10−6. Typically this takes less than m = 10 steps using

3For the sake of clarity, the index z in PZ (E) marks the probability distribution for the interaction with thefrequency-dependent environment, Zt (ν).

47


50000 node points on an interval ±5mV 4. With this extension of the P (E )-theory it is possible

to numerically calculate the spectral inelastic energy exchange probability PZ (E) between a

Josephson junction and an arbitrary frequency-dependent environment Zt (ν).

4.2.2 Thermal Broadening of the P (E ) Distribution due to Thermal Voltage Noiseu & Calculation of the Total P (E) Distribution

Finite temperature in the junction leads impose fluctuations u on the voltage drop across a

tunnel junction capacitance C J , as introduced in Sec. 3.2.3. These fluctuations act as another

source or drain of energy E = 2e(VJ +u), opening up another path for inelastic Cooper pair

tunneling. The corresponding PC (E ) distribution is of Gaussian shape and reads as [109, 107].

PC (E) =√

1

4πEC kB Texp

[ −E 2

4EC kB T

]. (4.17)

In comparison to the expression derived in reference [107] we neglected a spectral shift of the

PC (E) distribution due to Coulomb blockade, since the low DC impedance of the junction

leads in our experiment, RDC ¿ RQ , does suppress Coulomb blockade effects in the I (VJ )

curves.

For the experiments performed in this work these thermal fluctuations represent the only in-

elastic energy exchange path in addition to the interaction with the environmental impedance

Zt (ν). Hence, from these two distributions, PZ (E ) and PC (E ) respectively, the total probability

distribution can be calculated. Since both interactions are independent of each other, the total

Cooper pair tunneling probability can be obtained by convoluting PZ (E ) with PC (E ) [107, 109]:

P (E) =∫ −E0

E0

dE ′ PZ (E ′)PC (E −E ′). (4.18)

Figure 4.4 compares the pure probability distributions PZ (E) with the convoluted probability

distribution P (E). For the calculation of both P (E) distributions we employed the environ-

mental impedance function Z (ν) introduced in Sec. 4.3. As is expected from theory, the global

maximum of both distributions is located at E = 0eV [109]. However, the amplitude of PZ (E)

exceeds the amplitude of P (E) by more than a factor of 30. Additionally, the PZ (E) distribu-

tions peaks much sharper with a full width half maximum (FWHM) of σEZ ≤ 2µeV compared

to σEZ ≤ 80µeV. We conclude that the finite temperature, as the source of voltage fluctuation,

redistributes the spectral width of the P (E) distribution away from the zero voltage peak to

the area of finite voltage. This property is of great importance for the analysis of the measured

Cooper pair tunneling curves I (VJ ), as will be demonstrated in Sec. 4.5.

4The large energy interval guarantees that the strongly oscillating functions k(ν) and κ(ν) sufficiently decay tozero, otherwise the iterative calculation of PZ (E) would not converge.

48

4.3. The Environmental Impedance of a Typical STM Geometry

Figure 4.4 – Comparison of PZ (E ), which only contains the interaction with an environmentalimpedance, and the total probability distribution P (E).

4.3 The Environmental Impedance of a Typical STM Geometry

After deriving the necessary theoretical framework, the last piece in the puzzle, which is

required to model the experimental data, is a complex-valued analytic expression for the

environmental impedance Z (ν). Unfortunately, the technical constraints for operating an

STM do not allow for precise tailoring of the electric environment by means of nano- and

micro-lithography as is done by default in experiments on planar Josephson junctions, see e.g.

Refs. [39, 62]. Therefore, the frequency-dependent impedance of the environment has to be

estimated from considerations on the immediate junction environment.

First of all, the wires connecting the tunnel junction, i.e. STM tip and sample, to the mea-

surement circuit are long transmission lines, which, from the point of view of the junction,

can be considered infinitely long. Such transmission lines have a typical input impedance

of RDC = 377Ω, which determines the ohmic resistance Z (0) = RDC of the junction environ-

ment [61, 113]. The frequency-dependent part of Z (ν) can be approximated by investigating

the impedance distribution on the STM tip and sample assembly schematically shown in

Fig. 4.5(a). The bottom of the tip, which is 250µm thick and typically about 1 mm long, is

in electric contact with the tip holder, that forms a large plane. In contrast, the impedance

of the tunnel contact at the STM tip apex is high-ohmic, RN = 104 . . .106Ω, for which reason

we approximate the tip assembly as open-ended. Remarkably, the STM tip holder assembly

shares both the electric as well as the geometric properties of a monopole antenna connected

to a ground plane [113], as schematically displayed in Fig. 4.5(b). In this sense, the tip holder

forms the “antenna ground plane” (see Fig. 4.5(a)) 5. Therefore, the STM tip shares a similar

5To clarify the terminology, the tip holder acts effectively as the antenna ground plane although in the STM thetip holder is actually not connected to ground.

49


eigenfrequency spectrum with resonances at about

νn = (2n +1)× c/(4(l + l0)), (4.19)

where the integer n = 0,1,2, . . . denotes the nth eigenmode and l denotes the tip length with

an extension l0 corresponding to an electrical lengthening of the tip [114, 115]. A typical eigen-

frequency of a millimeter long STM tip is on the order of tens of GHz. For the measurement of

which the result is displayed in Fig. 4.2, for instance, a 1.7 mm long STM tip was used, whose

lowest eigenmode has the frequency ν0 = 30GHz. Converting the eigenfrequencies of this tip

into voltages, VJ = hν/(2e), we find the three lowest eigenmodes at approximately 60, 180 and

300µeV. These values are in good correspondence to the voltage positions of the resonant

current features we observed in Fig. 4.2, corroborating the validity of our geometric impedance

analysis6.

The schematic electric field pattern derived from this impedance analysis are also shown

for the three lowest eigenmodes ν0,1,2 in Fig. 4.5(c). The ground mode of the tip assembly

corresponds to a quarter wavelength, for which reason such an antenna geometry is also

called λ/4-monopole antenna or λ/4 resonator, respectively.

Since there is no analytic expression for a monopole antenna, we approximate the STM tip’s

impedance spectrum by an open-ended transmission line [113]. This is a valid approximation

since the open-ended transmission line and a monopole antenna share similar impedance

properties – a large impedance mismatch at one end and an impedance match at the other

end – which results in a similar excitation spectrum. However, this approximation neglects

any influence of the ground plane on the electromagnetic properties of the antenna. The

input impedance of the open-ended transmission line seen by the junction ZT (ν) reads as

follows 7 [107, 113]:

ZT (2πν) = Z (0)1+ i

α tan(πν2 ν0

)[

1−κ νν0

tan(πν2 ν0

)]+ iα

[κ νν0

+ tan(πν2 ν0

)] . (4.20)

Here, Z (0) denotes the input impedance of the tip, which for monopole antennas is typically

Z (0) = 36Ω [113]. This value is much smaller than the input impedance of the transmission

line RDC = 377Ω connecting the sample. Since the thermal resistor noise amplitude, causing

phase fluctuations, is dominated by the largest value of the ohmic resistance, we reduce

the number of parameters and set the input impedance of the tip Z (0) equal to the input

impedance of the transmission line RDC . The dimensionless parameter α is a measure for

the quality of the eigenmodes — smaller values correspond to eigenmodes of better quality

— and ν0 denotes the n = 0 eigenfrequency of the tip. The parameter κ = ν0Z0C J results

from the capacitive shunt of the junction by itself and, thereby, acts as an effective short-pass

6For this particular tip length the lowest eigenmode ν0 is hidden in the falling flank of the supercurrent peak.7In comparison to the pure environmental impedance Z (ν), the impedance ZT (ν) also contains the junction

capacitance C J , which shunts the junction itself (e.g. Ref. [109]).

50

4.4. Implementation of P (E)-Theory as Fitting Routine

filter, where Z0 ≈ 377Ω is the vacuum impedance. Figure 4.5(c) displays an example of the

calculated environmental impedance ZT (ν) of the STM tip assembly.

Figure 4.5 – (a) Sketch of the STM tip sample assembly. The red dot indicates a point of largeresistance R1 and the green dot indicates a point of small resistance R2. (b) Monopole antennaanalogue of the STM tip. The lowest eigenmodes of the tip holder assembly νn are displayedand indicated. (c) Complex impedance of a transmission line as a function of the frequency fora resonance frequency ν0 = 30GHz, a damping α= 0.75 and a short-pass constant κ= 0.005.

4.4 Implementation of P (E)-Theory as Fitting Routine

With a theoretical model and the environmental impedance in hand, we can fit the experi-

mentally measured I (VJ ) characteristics. However, the challenge in fitting such a curve using

P (E)-theory lies in the rather involved interplay of the different fitting parameters, as well

as in the large range of varying tunneling current amplitudes. Therefore, the experimental

parameters require more detailed consideration.

• We use the effective temperature of Teff = 40mK for the calculation of PC . The effective

temperature of the Josephson junction itself may also be higher, because we can not

perfectly shield it from stray photons. Although the STM head forms a closed metal

body, isolation ceramics for the wire feedthroughs are transparent for photons in the

MHz to GHz range. For the calculation of PZ (E) we therefore estimated the effective

temperature of the Josephson junction to T = 100mK, which results in a slightly better

fit, as compared to effective temperatures of either 50 or 150 mK, respectively.

• As discussed before, we can keep the ohmic resistance fixed at the input impedance of a

transmission line RDC = 377Ω.

• For typical experimental values of C J ≈ 1fF and ν0 ≈ 30GHz the short-pass parameter is

small, κ≈ 0.01. Since it, in general, has only a very small effect on the spectral properties

it will be kept constant for the further analysis at a value of κ= 0.01.

51


To account for the constant current background of the current amplifier, the equation for

the Cooper pair tunneling current, Eq. 4.10, is extended by a constant current offset, c1. The

gap-filling due to quasiparticle excitations (cf. Ch. 2, Sec. 2.2.3) is considered by adding a cubic

background c2 to the current equation. We find the final fitting function for the Cooper pair

tunneling current

I (VJ ) =πeE 2

J

ħ [P (2eVJ )−P (−2eVJ )]+ c1 + c2 ×V 3J . (4.21)

Incorporating these assumptions, we can fit the experimental I (VJ ) curve at GN = 0.27G0, as

shown in Fig. 4.6. The fit nicely reproduces both the supercurrent peak as well as the spectral

resonances and we can extract a Josephson coupling energy of E J = 46.60±0.12µeV. The

environmental impedance Z (ν) shows its lowest eigenmode frequency at ν0 = 29.47±0.20GHz.

This value matches the expected λ/4-resonance of the l = 1.7±0.1mm long tip used in this

experiment. The corresponding damping factor, α = 0.76± 0.01, has a rather large value,

which can be attributed to the intrinsically low quality factors of antennas 8. For the junction

capacitance, we find a value of C J = 2.16±0.04fF. This value is in good agreement with our

estimates in the experimental setup, which only comprises the junction capacitance between

tip and sample on the order of femtofarads.

We conclude that the I (VJ ) curves from our small capacitance tunnel junction, showing the

characteristics of sequential Cooper pair tunneling, can be nicely modeled by P (E )-theory with

reasonable parameters. Moreover, we are able to unambiguously determine an experimental

value of the Josephson coupling energy E J in a particular junction. In the following section we

will explore the conductance range of applicability for our implementation of P (E)-theory.

4.5 Testing the Range of Validity for P (E)-Theory

Comparing the relevant energy scales of our experiment in Fig. 4.1 at the beginning of this

chapter we found that we can tune the ratio of Josephson coupling energy E J versus capacitive

charging energy of the junction when changing the tunneling conductance GN via the tip-

sample distance. In this way, we can tune the Josephson junction from the DCB regime

E J ¿ EC at GN ¿G0 to a regime in which the Josephson coupling energy is comparable to the

charging energy E J ≈ EC at GN ≈ G0. However, P (E)-theory was derived exclusively for the

DCB regime so that it should fail to describe Cooper pair tunneling characteristics in this limit.

To address this interesting question, which has remained open so far, we will in this section

investigate the tunneling characteristics of the Josephson junction in our STM in the limit

GN →G0.

To this end, we measure the I (VJ ) curves for a wide range of tunneling conductance val-

8The purpose of antennas is to radiate the excitation into free space. Hence, in contrast to resonator cavities,the quality is strongly reduced.

52

4.5. Testing the Range of Validity for P (E)-Theory

Figure 4.6 – Example of a P (E)-fit to an I (VJ ) spectrum measured at GN /G0 = 0.27 and T =15mK. On the bottom, the real part of the environmental impedance that results from the fitis shown.

ues 0.05G0 ≤ GN ≤ 0.27G0 which are shown in Fig. 4.7(b). All curves exhibit a well-defined

supercurrent peak, symmetrically centered around zero bias, as well as spectral current res-

onances. Deviations are found in the background signal that originates from quasiparticle

excitations. To smaller values of the tunneling conductance the quasiparticle background

current increases and appears more pronounced, as compared to the Cooper pair tunneling

current features. Figure 4.7 also shows that by using P (E )-theory in the same fashion as before

we are able to nicely model all I (VJ ) curves along the entire conductance range that spans

almost three orders of magnitude. Concerning the fit parameter shown in Fig. 4.8, we find that

neither the junction capacitance C J nor the impedance properties, ν0 and α, depend on the

tunneling conductance. This result is certainly expected, since the spectral energy exchange

probability distribution does not depend on the tunneling conductance. Still, it corroborates

the consistency of our P (E) implementation and fitting routine as well as our capability to

correctly model I (VJ ) tunneling curves with P (E)-theory.

In our case, the regime GN →G0 is also of particular interest, since here the initial requirement

of P (E)-theory, E J ¿ EC , is not valid anymore (cf. Fig. 4.1). We should accordingly fail to

describe our experimental data with P (E)-theory. Additionally, Ingold et al. found that

the global condition E J ¿ EC is superimposed by another condition E J ×P (E) ¿ 1 [109].

This condition must hold for any applied bias voltage V = E/(2e) and for this reason, it

is much more stringent than E J ¿ EC . In order to test this hypothesis, we measured the

I (VJ ) curves for tunneling conductance values GN ≥ 0.59G0 of which three examples are

shown in Fig. 4.9. Using P (E)-theory as before, we were unable to properly fit any of these

spectra, which is to be expected since, at the measured conductance values, we find E J ≈ EC

(cf. Fig. 4.1). Nevertheless, we can up-scale a fitted current spectrum from experiments at

a lower conductance, GN = 0.27G0, making use of E J only being an independent scaling

53


Figure 4.7 – Experimental tunneling spectra I (VJ ) (color lines) and fit to data (black dashedlines) for indicated tunneling conductance values measured at T = 15mK.

54


factor. The up-scaled I (VJ ) curve nicely fits the spectral resonances at higher voltages, but

largely overestimates the supercurrent peak in all cases with increasing mismatch for higher

conductance values, as shown in Fig. 4.9, indicating the breakdown of P (E)-theory.

Figure 4.8 – Fitted values for the junction capacitance C J , the lowest eigenmode frequency ν0

and the damping α as a function of the normalized tunneling conductance GN /G0.

To better understand this observation, we investigated the product E J ×Pmax. For E J we use the

values found from upscaling the I (VJ ) curves. Pmax is the global maximum of the calculated

probability distribution P (E). It is found at zero voltage for the probability distribution of

the impedance PZ (E) as well as for the total convoluted probability distribution P (E) (see

Fig. 4.4) [109]. It can be seen that the broadening of the total P (E) due to the thermal voltage

noise u greatly reduces the maximum value of the P (E) distribution, as compared to PZ (E).

The dependence of E J ×Pmax on the tunneling conductance GN is shown in Fig. 4.10. For

G ≥ 0.59G0, we find E J ×Pmax ≥ 1 so that the required condition for P (E)-theory is “locally”

violated near zero voltage. This result perfectly explains our observation that P (E )-theory still

nicely models the spectral resonances at higher bias voltages, where E J×P (E À 0) ¿ 1, but fails

for the supercurrent peak close to zero bias, where P (E ) has its maximum and E J ×P (E ≈ 0) ≈ 1.

This result indicates that the Cooper pair tunneling current at low voltages is dominated by

the coherent tunneling of Cooper pairs whereas at higher voltages the sequential Cooper pair

55


Figure 4.9 – Experimental tunneling spectra I (VJ ) (color lines) and up-scaled fits (black dashedlines) for indicated tunneling conductance values measured at T = 15mK.

tunneling dominates the current. For this reason P (E )-theory fails to describe the entire I (VJ )

curve. This result is somewhat expected, since in the regime E J À EC the tunneling current is

sustained by the coherent tunneling of Cooper pairs and scales with the Josephson critical

current I0 or the Josephson coupling energy E J , respectively [31, 62]. Hence, as soon as Ingold’s

condition for sequential Cooper pair tunneling is violated, the Cooper pair current, which is

most likely dominated by coherent Cooper pair tunneling, will not scale as ∝ E 2J anymore but

rather scales as ∝ E J . For this reason, P (E)-theory will overestimate the Cooper pair current

amplitude around zero voltage, as is also reported by another group [116]. Our result also

signifies that the turnover from the sequential Cooper pair tunneling regime to the coherent

Cooper pair tunneling regime does not occur at a specific ratio EC /E J but continuously around

the turning point EC /E J = 1.

In order to discuss this physical interpretation we compare our experimental results with a

condition for coherent Cooper pair tunneling, which has been evaluated independently from

P (E)-theory by Falci et al. [110]. The authors investigated the effective action of a Josephson

junction employing the imaginary-time path-integral formalism [117, 118]. For a purely

ohmic environment Z (0) they found that coherent tunneling occurs as long as the condition

56


Figure 4.10 – Ingold’s condition E J ×Pmax calculated for PZ (E) and P (E) as a function of thenormalized conductance GN /G0.

γ = E J /(EC ×α1/2) > 1 holds, where α is defined as α = RQ /Z (0). Fig. 4.11 displays γ as a

function of the normalized tunneling conductance. For EC and E J we used the values found

from the fit and as before, we set Z (0) = RDC = 377Ω. In fact, for the entire conductance range

we find γ< 1 so that tunneling current is still carried by the sequential tunneling of Cooper

pairs, as we found from our analysis using P (E)-theory. For GN /G0 > 1, γ approaches unity,

which indicates the turnover to the coherent tunneling regime in the framework of this theory,

similar to the condition EC À E J . However, both conditions γ> 1 and EC À E J , respectively,

are global conditions and it requires the more stringent, local requirement E J ×P (E) ¿ 1 in

order to properly explain our experimental results.

Figure 4.11 – The parameter γ as a function of the normalized conductance GN /G0 [110].

57


Moreover, the analysis of the P (E) function reveals the significance of the thermal voltage

noise as a spectral broadening mechanism (cf. Sec. 3.2.3 and 4.2.2). We cannot fit the experi-

mental data by P (E)-theory without considering the capacitive noise broadening. While the

probability distribution in the convoluted P (E)-function is broadened and has some spectral

weight at higher voltages, the PZ (E) distribution – only containing the interaction with the

frequency-dependent environment – sharply peaks at VJ = 0 (see Fig. 4.4). For this reason,

the required condition E J ×PZ ,max ¿ 1 is violated for almost the entire conductance range as

shown in Fig. 4.10. This is in agreement with theory, which predicts failure of P (E )-theory for a

tunnel junction that is operated in a low impedance environment, i. e. RDC ¿ 1/RQ . However,

including the capacitive noise broadening [109, 107] is necessary to correctly describe the

experimentally observed sequential Cooper pair tunneling, because we still work at non-zero

temperatures. It reduces the Pmax values and results in an overall consistent picture between

experiment and theory as well as the range of validity for P (E)-theory.

4.6 The Josephson Critical Current in the Dynamical Coulomb Block-

ade Regime

In the last section, we demonstrated that we can model Cooper pair tunneling characteristics

using P (E)-theory over a large range of tunneling conductance values and that P (E)-theory

is valid across this range. We can therefore unambiguously determine the local value of the

Josephson coupling energy E J . However, in order to access the local superconducting order

parameter ∆ via Josephson STM, it is necessary to determine the local value of the Josephson

critical current I0, simply related to E J via natural constants, I0 = (2e/ħ)E J . However, the well-

known formulation of the Josephson critical current derived by Ambegaokar and Baratoff [8],

which relates I0 to ∆, was derived at exactly zero voltage, where the tunneling current is purely

carried by the coherent tunneling of Cooper pairs. Such a case corresponds to a large planar

junction of large capacity and small tunneling resistance, in which EC and both quantum

and thermal fluctuations can be disregarded. As we have demonstrated in the previous

sections, STM experiments are commonly operated in the dynamical Coulomb blockade

regime, in which the Cooper pair tunneling current is dominated by the sequential tunneling

of Cooper pairs. For this reason it is a priori not clear that the experimentally determined E J

values will result in the Josephson critical current as defined by Ambegaokar and Baratoff. To

quantitatively compare the experimentally found values for the critical current with the critical

current calculated from the AB formula, we again write this formula for two superconductors

with unequal order parameters ∆1,2, where ∆1 >∆2, and zero temperature 9 (cf. Eq. 2.21 in

Sec. 2.2.5)[8]:

I0 =∆2GN K

√√√√1− ∆

22

∆21

. (4.22)

9In good approximation we can neglect the temperature.

58

4.6. The Josephson Critical Current in the Dynamical Coulomb Blockade Regime

To gain access to the order parameters ∆1 and ∆2, we measured the quasiparticle excitation

spectrum of the tunnel junction in our STM, which is shown in Fig. 4.12. In the lower panel the

differential conductance proportional to the LDOS is displayed. The upper panel shows the

second derivative, which enhances the positions of the Andreev reflections. The differential

conductance shows two peaks at ∆1 +∆2 = 1210µeV as well as in-gap states. We find an

Andreev reflection that is associated with the sample gap ∆1 at eV ≈ 760µeV [50, 53]. Due to

lifetime effects of Cooper pairs in the STM tip, the spectral features in the d I /dV spectrum

are broadened [54]. Still, the gap values can be extracted from the d 2I /dV 2-spectrum at

∆1 = 760±70µeV and ∆2 = 450±20µeV.

Figure 4.12 – First derivative d I /dVJ of the tunneling current (bottom) and second derivatived 2I /dV 2

J (top), respectively.

Inserting these values along with the tunneling conductance GN into the AB formula, we

can plot the corresponding critical currents in Figure 4.13 as a function of the normalized

tunneling conductance GN /G0. The critical currents from the P (E)-fits and the AB formula

match within error bars over the entire range of tunneling conductance where GN ≤ 0.27G0.

On the bottom, Fig. 4.13 also displays the relative deviation δI0 between calculation and

experiment, δI0 = 100× |I0,exp − I0,calc|/I0,calc. In all cases, δI0 is smaller than 10 % and we

determine its standard deviation to σδI0 ≤ 5%. The agreement between the experimental

and the calculated critical current is a remarkable observation, because the experimental

values were determined from sequential Cooper pair tunneling characteristics, whereas the AB

formula was derived for coherent Cooper pair tunneling. Hence, our findings suggest that the

critical current I0 represents a coupling strength between the overlapping pair wavefunctions

and that this coupling strength is independent of the actual tunneling process – a question that

remained open in the original work of Ambegaokar and Baratoff [8]. In this sense, the spectral

weight of the Cooper pair tunneling current in the DCB regime, E J ¿ EC , is distributed at

finite voltages, while it peaks at zero voltage for Josephson junctions of negligible capacitive

charging energy, such that E J À EC .

59


In Fig. 4.13 we also compare the critical current values of the up-scaled fits with the values

calculated using the AB formula. Here, the deviation is much larger, δI ′0 ≤ 41%, which, however,

is not too surprising. The large tunneling conductance in these experiments, GN /G0 ≥ 0.59,

clearly violates the weak coupling limit, i.e. GN ¿G0, for the perturbative calculation of the

Cooper pair current through the Josephson junction (cf. Sec. 2.2.5) [7, 8]. Instead, it could be

experimentally shown, that the Cooper pair current through transparent Josephson contacts,

i.e. GN ≈G0, results from a population imbalance of Andreev bound states on each side of the

atomic contact altering the sinusoidal phase relationship of the Josephson effect in the weak

coupling limit [119]. As a consequence, the critical Josephson current in this limit deviates

from the result obtained by Ambegaokar and Baratoff, as we could also find indications for

in our experimental data. Moreover, in this so-called Andreev bound state model the Cooper

pair current is transmitted through the i number of conductance channels of the Josephson

contact, whereby each of them has a specific, yet arbitrary, transmission τi [120, 121]. For this

reason, this theory can be applied to a variety of Josephson contacts ranging from the tunnel

junction, as studied in this work, to ballistic contacts. However, we only recently started to

investigate our tunnel contact configuration in more detail by using a Hamiltonian approach

as given in Ref. [122]. Until now, we lack knowledge on the precise number of conductance

channels as well as their respective transmission and for this reason, the discussion on the

critical Josephson current in the limit GN →G0 has to remain on this qualitative level.

It should be noted that the formulation of the Josephson critical current as derived by Am-

begaokar and Baratoff [8] corresponds to the limit of many poorly transmitting channels, i.e.

i →∞ and τi ¿ 1, of the Andreev bound state model [123]. Hence, it should only be possible

to reach the Ambegaokar-Baratoff critical current, if the tunnel contact consists of a sufficiently

large number of conductance channels. This requirement is certainly met for planar tunnel

junctions of large junction area, as experimentally demonstrated by Steinbach et al. [62], but

not a priori fulfilled for the atomic scale tunnel junction in an STM. In our case this turns

out as an interesting observation. We found great agreement between our experimentally

determined critical current values and the calculated values using the AB formula, which

indicates that the tunnel junction in an STM in fact consists of a large number of conductance

channels. This observation is in stark contradiction to the commonly-used hypothesis that

in STM experiments only one poorly transmitting conductance channel contributes to the

tunneling current [75]. Since the tunnel contact configuration in an STM recently attracted

an increasing amount of interest [124, 125], we will address this interesting problem in future

work.

4.7 Conclusion

In summary, we have investigated the I (VJ ) tunneling characteristics of a voltage-biased

Josephson junction in the DCB regime for the purpose of JSTM. We showed that using P (E)-

theory, we are able to describe the experimental data with high accuracy and extract the

Josephson coupling energy E J . We also observed indications for the continuous transition

60

4.7. Conclusion

Figure 4.13 – Critical current I0 as found from the P (E)-fit to the experimental data andcalculated using the AB formula (top). Relative deviation between fitted and calculated I0

values (bottom).

from the regime of pure sequential Cooper pair tunneling to a regime, in which both sequential

Cooper pair tunneling and coherent tunneling sustain the tunneling current in the same

spectrum but at different energies. In this way, we could experimentally determine the range

of validity for P (E)-theory and, at the end, define a parameter range for JSTM to work best.

Furthermore, we found that the Josephson critical current values I0 = 2e/ħE J , as calculated

from the experimental E J values, are equivalent to the theoretical values calculated by using

the Ambegaokar-Baratoff formula [8]. The DCB regime, which is predominantly accessible in

STM, can therefore be used to directly determine unambiguous values of the critical Josephson

current I0. This result represents the fundamental step towards the implementation of JSTM

as a versatile spectroscopic tool for studying superconductor physics on the atomic scale

[14, 16, 17, 18].

61

5 The AC Josephson Effect as a GHzSource on the Atomic Scale

The content of this chapter is published as a letter in Applied Physics Letters 106, 013109 (2015)

entitled A nanoscale gigahertz source realized with Josephson scanning tunneling microscopy

by the authors Jäck, B., Eltschka, M., Assig, M., Hardock, A., Etzkorn, M., Ast, C.R. and Klaus

Kern.

5.1 Introduction

Within the last chapter we showed that using P (E)-theory it is possible to describe the I (V )-

characteristics of a Josephson junction operated in the dynamical Coulomb blockade regime

(DCB). Our experimental data reveals the high sensitivity of the Josephson junction in our STM

to its immediate, frequency-dependent environment Z (ν). In particular, this sensitivity en-

ables inelastic Cooper pair tunneling via energy exchange processes with the electromagnetic

environment Z (ν). From a simple geometric analysis we found that the frequency-dependent

impedance Z (ν) of a typical STM geometry, such as that shown in Fig. 5.1, is dominated by the

tip holder sample geometry and shares the electric properties of a λ/4-monopole antenna.

In this chapter, we study the electromagnetic properties of the λ/4-monopole antenna and its

interaction with the Josephson junction in more detail. On the one hand, the combination

Figure 5.1 – Conceptual sketch of the STM tip holder sample geometry and the three lowesttip eigenmodes.

63

Chapter 5. The AC Josephson Effect as a GHz Source on the Atomic Scale

of a Josephson junction and a resonator in its immediate environment represents a valuable

tool for studying the photon emission process from tunneling and has recently attracted

an increasing amount of research interest [105, 126, 127]. In particular, possible nonlinear

interactions of an excited resonator and a Josephson junction are predicted by theory [106].

On the other hand, combining the junction’s sensitivity to GHz signals with a resonator of

GHz resonance frequencies could allow us to realize a spectroscopy tool for GHz signals

at the nano-scale, a concept that has already been successfully demonstrated on planar

junction geometries [22, 23]. It is of great interest to introduce this frequency range into

the vivid research field of nano scale science, because it would facilitate a new approach

for investigations on individual molecular magnets [24] or the physics of individual nuclear

as well as electronic spins, for instance [128, 129]. However, generating and guiding GHz

signals from outside the experimental setup into the tunnel junction requires adaptation to

the experimental conditions of STM, adding another level of complexity. All recent realizations

of an STM employing high frequency radiation (HF) feed externally generated HF-signals into

the junction by means of optical guiding or waveguides [130, 131]. As we will show in the

following, the fundamental aspect of this study lies in circumventing the external feed and,

instead, realizing both GHz signal generation as well as detection right in the tunnel junction.

To this end, we combine the AC Josephson effect, acting as a perfect voltage VJ to frequency ν

converter through the relation hν= 2eVJ , with the tip resonator that provides the necessary

electromagnetic eigenmode spectrum. This unique combination, along with the junction’s

sensitivity to its immediate electromagnetic environment and the local probe capabilities of

an STM, could make the AC Josephson STM an ideal broadband atomic scale spectroscopy tool

operating under UHV conditions.

5.2 Tuning the Tip Eigenmodes

In Section 4.3 we found, from a simple analysis of the geometric impedance distribution, that

the tip holder assembly shares the electric properties of a λ/4-monopole antenna. Since the

eigenmode energy of such a monopole antenna reciprocally depends on the antenna length

l (cf. Eq. 4.19), we should be able to tune the position VJ = hν/(2e) of the resonance current

peaks in the nanoscale Josephson junction by changing the length l of the STM tip (cf. Fig. 5.1).

In order to test this hypothesis, we measure I (VJ ) curves for different tip lengths ranging from

l = 2.7±0.1mm to l = 0.7±0.1mm, which are shown in Fig. 5.2. It can be clearly seen from

the spectra that the current resonance peaks move to higher energies as the tip length l is

reduced. The highest measured current peak that we observe for an l = 0.7±0.1mm long

STM tip corresponds to a resonance frequency of the lowest eigenmode exceeding 200 GHz.

For longer tips we cannot, however, resolve a footprint of the n = 0 mode in the I (VJ ) curves,

which in these cases is part of the falling flank of the supercurrent peak close to zero voltage.

Still, the characteristics of the visible current resonances unambiguously reflect the typical

eigenmode spacing of a monopole antenna (cf. Eq. 4.19).

64

5.2. Tuning the Tip Eigenmodes

Figure 5.2 – I (VJ ) curves, normalized to the supercurrent peak amplitude ISC P , for experimentsat a conductance GN = 0.1G0 using STM tips of different length l = 2.7±0.1mm, l = 1.7±0.1mm and l = 0.7±0.1mm, from bottom to top. The tip eigenmodes νn are indicated at thecorresponding current resonance.

By comparing the I (VJ ) curves in Fig. 5.2, we also find that the amplitude of the current

resonances is enhanced when the tip length is reduced. We can explain this observation in

a descriptive fashion if we consider the current distribution oscillating along the antenna.

Similar to the well-known dipole antenna, where the current oscillates between the two edges

of the antenna, the current on the monopole antenna oscillates between the antenna apex

and the outer edge of the ground plane. In other words, the ground plane constitutes the

monopole antenna’s second half and acts as an electric counter weight [113]. As a general rule,

the ground plane radius r should not be smaller than the antenna length.

Considering the various tip lengths we used in our experiment, we find that only the l =0.7±0.1mm STM tip matches the rule r /l ≈ 1 and, moreover, that for this particular tip length

the current resonances have the largest amplitude. To further investigate the relation between

antenna ground plane size and antenna properties, we change the tip holder surface area,

which is shown in Fig. 5.3(a). While the antenna length, l = 0.7± 0.1mm, stays the same,

we increase the surface area by a factor of three. Figure 5.3(b) displays the measured I (VJ )

curves for these two different tip holder geometries. For the larger surface area A1, the peak

amplitude more than doubles compared to the smaller surface A2, where it almost reaches the

supercurrent peak amplitude. In contrast, the voltage position of the current resonance at the

corresponding voltage VJ = hν0/2e remains unaffected. Hence, we experimentally observe

that changing both the length of the tip and the size of the tip-holder strongly changes the

electromagnetic properties of the tip holder assembly, i.e. of the monopole antenna.

To quantitatively analyze the relation between the geometry and the electromagnetic prop-

65


Figure 5.3 – (a) Sketch of the two different tip holder surface geometries used for the experi-ments (The dimensions are indicated). (b) I (VJ ) curves, normalized to the supercurrent peak,for experiments with different tip holder surfaces A1 and A2, both measured at a conductanceGN = 0.1G0.

erties of the tip holder assembly we employ the particularities of P (E)-theory, namely, that

the current features – expressing the spectral energy exchange probability P (E) with the envi-

ronment – directly relate to the impedance Z (ν) of the tip holder assembly. In this sense, we

perform an impedance spectroscopy by using the sequential Cooper pair tunneling current as

a probe [39]. To this end, we fit the experimental I (VJ ) curves with P (E)-theory in the same

fashion as introduced in the last chapter (cf. Ch. 4, Sec. 4.4). From the fits we can extract the

properties of the tip impedance Z (ν) which are its lowest mode frequency ν0 and the damping

parameter α (cf.Eq. 4.20). In Fig. 5.4(a) we plot the frequency of the lowest eigenmode ν0

for the different measured tips and tip holder shapes as a function of the tip length l . Their

values nicely follow the inverse proportionality to the tip length as found in Ch. 4, Sec. 4.3 from

a simple geometric impedance analysis. Using the relation between the tip length and the

lowest eigenmode frequency,

ν0 = c/(4(l + l0)), (5.1)

we can fit the dependence of ν0 on the tip length l and extract an electrical lengthening of

l0 = 0.46±0.04mm. This rather large value, even exceeding the shortest tip length, is not

surprising, because lengthening effects are predominantly observed for antennas of non-

negligible thickness, as is the case with our STM tips [115, 113]: For the shortest tip, l = 300µm,

we are in the extreme limit in which the antenna length is even comparable to its thickness

d = 250µm.

Quantitatively analyzing the relation between the fittedα values and the geometric parameters

remains difficult, because the impact of the ground plane geometry on the electromagnetic

properties of the antenna is rather complex [115]. Hence, it is not possible to derive an

analytical expression for the antenna impedance Z (ν) that connects to the geometry, for

66

5.2. Tuning the Tip Eigenmodes

Figure 5.4 – (a) Lowest eigenmode frequency ν0 as a function of the tip length l for both tipholder geometries A1 and A2, respectively, found from the P (E)-fits to the I (VJ ) curves andthe FIM simulations. (b) Damping parameter α as a function of the tip length l found from theP (E)-fits to the I (VJ ) curves.

which reason the discussion ofα has to remain on a qualitative basis at this point. Figure 4.4(b)

displays the damping α as a function of the tip length l that we found from the fits. These

large values reflect the intrinsically low quality common to antennas that are constructed for

dissipating power into free space. Moreover, unlike real antennas that are capacitively coupled

to their power feed, the STM tip in our case is in ohmic contact to the measurement circuit.

This link favors power dissipation along the wires and results in the observed large damping.

Figure 4.4(b) also compares the fitted damping for the two different tip holder geometries.

For the large tip holder surface A1 we find a strongly reduced damping compared to the

small surface tip holder A2. This result reflects the ground plane’s relevance as the antenna’s

“second half". The slight increase found inα for shorter tip lengths with the tip holder A1 could

originate from the discussed finite thickness of the antenna reducing the resonance quality.

67


5.3 Numerical Simulations of the Environmental Impedance

In a next step, we aim to corroborate and complement our knowledge on the electromagnetic

properties of the STM geometry by means of Finite Integral Method (FIM) simulations using the

CST Microwave Studio [132]. The simulations were performed in collaboration with Andreas

Hardock from the Institute for Theoretical Electrical Engineering at the TU Hamburg-Harburg.

To this end we create a simplified 3D model of the STM head geometry, of which cross sections

are displayed in Fig. 5.5 (cf. Ch.3, Fig.3.2). In comparison to the real STM head geometry, we

reduce the complex structure to the relevant parts in the junction vicinity, which is a necessary

step for a successful simulation, as the wide spread of aspect ratios along the real geometry

results in diverging simulation times. However, this reduction to the relevant structure could

result in deviations between the real and simulated impedance Z (ν), in particular in the limit

of lower frequencies where large geometric aspect ratios are of importance.

Figure 5.5 – (a) Top view of the geometric structure used for the FIM simulations. (b) Crosssection of the same structure. The excitation port and the small gap between tip apex andsample surface are indicated. All relevant dimensions are labeled.

To probe the electromagnetic properties of this reduced structure, we apply an excitation pulse

at the tip bottom and record the scattering spectrum of this excitation along the geometric

structure as the scattering parameter S11(ν). As a pulse, we use a Dirac-shape pulse in the time

domaine, which has an amplitude of V = 1 V. In order to simplify the analysis of S11(ν), we

neglected reflections from the enclosing structure. The impedance Z (ν) of the tip holder as-

sembly can be directly calculated from the scattering parameter using the network microwave

conversion technique [133]. For a one port network, this conversion is simply given by

Z (ν) = Z (0)1+S11(ν)

1−S11(ν), (5.2)

where Z (0) denotes the DC impedance of the structure.

Fig. 5.6(a) and (b) display the scattering parameter S11(ν) as well as the converted impedance

spectrum Z (ν), which result from the FIM simulation of a l = 1.7mm long STM tip on the

small area tip holder A2. They feature four resonances on the sampled frequency range, which

68

5.3. Numerical Simulations of the Environmental Impedance

are indicated by ν0,1,2 and νX . The frequency values of the ν0,1,2 modes constitute the typical

eigenmode pattern of a λ/4-monopole antenna, namely that the lowest eigenmode ν0 has

approximately half the frequency of the spacing of the higher modes ν0 ≈ 0.5(nk −nk−1) and

k ≥ 2 (cf. Eq. 4.19). We can observe a similar eigenmode pattern in the voltage positions of the

current resonances in experimental I (VJ )-curves (cf. e.g. Fig. 5.2). However, there are distinct

differences between the simulated impedance spectrum and our experimental results. First,

we never observe a footprint of the νX ≈ 25GHz mode in the experimental I (VJ ) curves (cf.

Fig. 5.6(b) and 5.2). Since the corresponding current resonance would occur at voltages of

around 50µeV it could be masked by the much larger supercurrent peak occurring at similar

voltages. Second, the amplitude of the eigenmodes ν0,1,2 rapidly decays to higher modes n > 0.

When comparing to experimental I (VJ ) curves, e.g. Fig. 5.2, the current resonances also exhibit

such a decay to higher modes. However, also the response of the Josephson junction to the

environment decays as 1/ν [37, 38]. For this reason it is difficult to unambiguously assign the

decay in the current resonances to a decay of the resonance amplitudes in Z (ν). Our analysis

of the experimental data based on P (E )-theory using a transmission line impedance does not

include such a decay, cf. Eq. 4.20 and Fig. 4.5(c). Therefore, our analysis could contain an error

in the relative distribution of the spectral weight. On the one hand, a more precise treatment

of Z (ν) is certainly desirable to reduce this error and will be part of future work. In particular,

using numerically simulated impedance functions as an input parameter for P (E)-theory

could be of particular interest here. On the other hand, the known error in Z (ν) should not

hamper a quantitative analysis of the Cooper pair tunneling curves in view of JSTM, which

was indeed demonstrated by extracting the correct values of the Josephson critical current in

Ch. 4, Sec. 4.6.

Figure 5.6 – (a) Scattering parameter S11(ν) of the simulated geometry, as shown in Fig. 5.5(l = 1.7mm, A2), with four indicated resonances, ν0, 1, 2 and νX, respectively. (b) Real- andimaginary part of the impedance Z (ν), as obtained from the simulated scattering parameterspectrum.

Analyzing the experimental Cooper pair tunneling characteristics using P (E)-theory, as de-

scribed in the previous section, we observed a strong dependence of the environmental

impedance Z (ν) on the STM geometry. To reproduce this observation we simulate the electro-

69


dynamic properties of the corresponding simplified geometries, i.e. different tip lengths l and

tip holder surfaces A, in the same fashion as before and determine the simulated impedance

function Z (ν, l , A). From these impedance functions we extract the respective lowest eigen-

mode frequency ν0 and plot them as a function of the tip length in Fig. 5.4(a). We find largely

good agreement between these simulated values and the values determined from experi-

mental I (VJ ) curves using P (E)-theory, also concerning the effect of electrical lengthening

[113, 114].

In the last step, we investigate the electric near field pattern of the resonance modes that can

be found from the FIM simulations. Figure 5.7 displays the electric field pattern on a cross

section of the simplified structure, focusing on the tip holder sample area. Shown are the

absolute values of the electric field at the frequencies of all resonances ν0,1,2 and νX , that are

found in the impedance spectrum (cf. Fig. 5.6(b)). For the eigenmodes ν0,1,2, these patterns

are in great agreement with the simplified electric field pattern in Fig. 4.5(b), which we derived

from a basic geometric impedance distribution analysis. We further find the field strength

to be highest at the tip apex and, due to the confined geometry, we estimate the near field at

the tip apex, i.e. in the tunnel junction, to be much larger than the field strength found from

simulations 1. By reciprocity, the antenna is most susceptible to absorb radiation at this point,

thereby corroborating the observed strong coupling to the Josephson junction (cf. Sec. 4.2) 2.

In contrast to the tip eigenmodes, the field pattern of the mode νX appears like a resonance

mode that is hosted by the enclosing surfaces between tip-holder and sample. It can also be

seen that the antenna ground plane, i.e. the tip holder, shows enhanced field strength for

every mode, which illustrates its relevance and the impact observed in our experiments (cf.

Fig. 5.4(b)).

In conclusion, the presented simulation results on the electrodynamics properties of the STM

head confirm and extend our results on Z (ν) that we obtained by means of I (VJ ) tunneling

spectroscopy as well as from a simple geometric impedance analysis shown in Ch. 4. We

conclude that the impedance of a typical STM geometry is strongly frequency-dependent and

exhibits characteristic properties of a λ/4-monopole antenna, such as a defined eigenmode

spectrum with the λ/4 resonance as the lowest eigenmode. First, this result is of great interest

for ourselves because it clearly characterizes the environment of the Josephson junction in

our STM. Second, our results on the STM head impedance could provide useful information

for other recently implemented STM setups that involve HF signals and operate in the time

domain [131, 134, 135].

In the following section, we will discuss perspective applications for the combination of a

resonator structure and a Josephson junction in an STM.

1Even for the simplified geometry, the largely varying aspect ratios did not allow a detailed study of the fielddistribution in the tunnel junction in these first FIM simulations. However, with the present knowledge obtainedgranting a precise refinement of the simulated structure and mesh, the near field properties in the tunnel junctionwill be investigated in future simulations.

2The current amplitude at a given voltage VJ directly relates to the energy exchange probability at the corre-sponding energy P (E = 2eVJ ), which in our case is dominated by the tip holder assembly impedance.

70

5.4. Features and Perspectives of the AC Josephson STM

Figure 5.7 – Simulated electric field pattern for the three lowest tip eigenmodes ν0,1,2 as well asthe additional geometrical mode νx of a l = 1.7mm long tip.

5.4 Features and Perspectives of the AC Josephson STM

In view of the resonator properties of the STM tip and the maximum electric field at the

tip apex, we look again at the inelastic tunneling process of Cooper pairs in more detail:

The energy hν of the emitted photons can be directly tuned by the applied bias voltage VJ

(2eVJ = hν). The tip eigenmode spectrum enhances the tunneling probability of Cooper pairs

at specific eigenmode energies hνn . This means that the tip eigenmode spectrum facilitates

the creation of a photon field in the tunnel junction. Remarkably, while the tunneling process

and the photon generation are localized in the nanoscale tunnel contact of the STM, the

macroscopic resonator, i.e. the STM tip holder assembly, necessary to enhance the tunneling

process, can be manipulated on the millimeter scale. In conjunction with the unity quantum

yield of photon generation, these two different length scales should facilitate a high photon

flux φ localized at the tunnel contact. This flux can be directly calculated from the resonant

current peak amplitude, since the quantum efficiency of the photon generation process is

unity, i.e. every inelastic Cooper pair tunneling process generates one photon of energy

hν= 2eVJ . For a measurement with an l = 0.7±0.1mm long tip on the large tip holder surface

A1 shown in Fig. 5.8, we find a peak amplitude for the ν0 mode of IM ≈ 350pA at a tunneling

conductance of GN = 0.12G0. This value directly translates into a photon number per second

of np ≈ 0.9× 109 s−1. If we assume, in a simplified picture, that the photons are emitted in a

cube of edge length l , we can estimate the approximate number of photons crossing a surface

plane of this cube φ= np/(6× l 2). However, it is currently an unresolved question whether the

photons are emitted during the tunneling process through the atomic scale tunnel contact,

l ≈ 100,pm, or after tunneling within the coherence length of the superconductor, l ≈ 101 nm

[93]. Still, from these values we can estimate upper and lower boundaries of the photon

flux to 1020 ≤ φ ≤ 1024 cm-2 s-1. These flux values are well within the range of flux applied

in conventional laser spectroscopy methods, hence we can underline the potential of the

resonator Josephson junction combination in our STM to be employed as a high-frequency

source for spectroscopic applications.

However, it should also be noted that a successful application of the AC Josephson STM for

spectrometry does actually not require a resonator for the creation of an additional photon

field inside the tunnel junction that could interact with nanoscale samples. In principle, it only

71


requires the sensitivity of the Josephson junction to its frequency-dependent environment

Z (ν). In detail, following P (E)-theory (cf. Ch.4), any electromagnetic transition that provides

a possibility for Cooper pairs to exchange the energy portion hν= 2eVJ alters the total envi-

ronmental impedance spectrum Z (ν). Accordingly, also a nanoscale sample inside the tunnel

contact that exhibits such an electromagnetic transition, e.g. the spin-flip process of a mag-

netic moment, should alter the measured Cooper pair tunneling current at the corresponding

voltage VJ = hν/(2e). Hence, the AC Josephson STM should be capable of probing a variety of

nano-scale samples, the choice of which is only limited by the spectrometer’s spectral range

and energy resolution, i.e. spectroscopic line width.

The accessible frequency range is limited by the superconducting order parameter ∆ of the

electrode material, since the photon generating process involves tunneling Cooper pairs.

For our current setup using vanadium as the electrode material, the maximum frequency of

approximately 250 GHz is limited by the maximum value of the tip order parameter∆≈ 500µeV.

Still, this range could be extended into the THz range by choosing materials with larger ∆,

such as MgB2 [136]. However, in that case the junction capacitance and wire resistance

have to be adapted to the higher frequency range. The line width of the AC Josephson STM

is predominantly limited by the voltage noise u due to thermal charge fluctuations on the

shunting capacitance, which effectively broadens all spectral features (cf. Ch. 4, Sec. 4.2.2).

Hence, the lowest temperatures are necessary to minimize the line width. For our current

setup we estimate voltage fluctuations of about

√u2 ≤ 12µeV [35]. Inserting a larger capacitive

shunt will strongly decrease the line width, but in turn also decrease the junction sensitivity

to large frequencies, so that the right shunt needs to be chosen in view of the perspective

application.

Figure 5.8 – Determination of both the maximum resonance current IM and the FWHM σ ofthe lowest eigenmode ν0 for a measurement with an l = 0.7mm long STM tip on top of the tipholder A1.

Another interesting aspect is the lifetime of an excitation in the resonator, i.e. the tip antenna,

72

5.5. Conclusion

after an inelastic Cooper pair tunneling event. For lifetimes larger than the average time

between two subsequent inelastic Cooper pair tunneling events, the excited resonator could

allow us to study nonlinear Cooper pair photon interactions predicted by theory [106]. From

the current resonance in Fig. 5.8 we can make a simple estimate on this lifetime τ1, which

can be approximated via the oscillation period of an eigenmode along the tip antenna, t ,

and the quality factor of the corresponding eigenmode τ1 = t ×Q. Here, the quality factor

denotes the number of oscillation periods for a resonator excitation before the resonator

relaxes back into its ground state. We can calculate the oscillation period as the time that

an excitation needs to travel from the tip apex to the tip-holder edge and back to the apex

t = 2(l +d/2)/c0. For the l = 0.7mm long tip on the large tip-holder of diameter d = 3.2mm

used for the experiment in Fig. 5.8 we find t ≈ 44ps. Furthermore, we can simply approximate

an upper limit of the resonator quality from the FWHM value σ and voltage position VJ of

the current peak to Q = VJ /σ ≈ 3. These values result in an average excitation lifetime for

the resonator of τ1 ≈ 0.13ns. From the amplitude of the current resonance in Fig. 5.8 we can

also calculate the average time between two subsequent resonator excitations as the average

time between two inelastic Cooper pair tunneling events, and we find τ2 = 2e/IS ≈ 0.9ns.

Remarkably, the average excitation lifetime of the resonator, τ1, is smaller but still on the same

order of magnitude as the average time between two excitation events. Hence, experiments

on nonlinear Cooper pair photon interactions [106] are in reach and, for instance, could be

realized by tuning the geometry of the tip antenna in order to increase the eigenmode quality

(cf. Ch. 7).

5.5 Conclusion

To conclude, in this chapter we investigated the electrodynamic properties of the Josephson

junction environment, Z (ν). Combining voltage-biased experiments on the Josephson junc-

tion in our STM with FIM Simulations on the electrodynamic properties of the STM head,

we find that the impedance Z (ν) of a typical STM geometry is strongly frequency-dependent

and that it is similar to the impedance of a λ/4 monopole antenna. With the sensitivity to

HF-signals we demonstrate a first realization of the AC Josephson STM spectrometer and

show that spectrometry of HF signals is, in principle, possible with the microscopic Josephson

junction in an STM, a concept that has only, to date, been successfully demonstrated in planar

junction geometries [22, 23, 137, 138]. The spectral range of this spectrometer is only limited

to larger frequencies by the superconducting order parameter ∆, whereas the line width of the

spectrometer is dominated by thermal voltage fluctuations on the junction capacitance.

73

6 Phase Dynamics of an Atomic ScaleJosephson Junction

6.1 Introduction

In Chapter 4, experimental results indicated that for larger values of the tunneling conductance

GN ≈G0 the tunneling current through the Josephson junction in our STM is carried by an

increasing amount of coherent Cooper pair tunneling, as opposed to the sequential tunneling

of Cooper pairs in the DCB regime. The observation of coherent Cooper pair tunneling is

associated with a bound state of the junction phase inside a washboard potential minimum,

which facilitates the observation of the DC Josephson effect. In fact, we are able to observe the

DC Josephson effect when we apply a bias current to the Josephson junction in our STM as, for

example, shown in Fig. 6.1(a) and (b). As we will see later, this even works for quite a large range

of tunneling conductance values GN < G0. Figure 6.1(b) also shows that in our experiment

the I (VJ ) curves are strongly hysteretic and, even more remarkably, that the current-biased

and voltage-biased I (VJ ) curves perfectly match and only deviate in the regions of negative

differential conductance. By being able to access features of both sequential and coherent

Figure 6.1 – (a) Large scale comparison of the I (VJ ) curves for voltage-biased and current-biased experiments. (b) Magnification of the in-gap region that displays the sequential Cooperpair tunneling characteristics discussed in Ch. 4 for the voltage-biased experiment as well asthe characteristic hysteretic I (VJ ) characteristics for the current-biased experiment.

75

Chapter 6. Phase Dynamics of an Atomic Scale Josephson Junction

tunneling in one spectrum as well as the existence of the DC Josephson effect in the current-

biased I (VJ ) curves tells us that our Josephson junction is operated in a quantum mechanical

regime (cf. Ch. 2, Sec. 2.5). In this regime, E J ≈ EC , neither the junction phase φ nor the

charge q are perfectly defined quantum numbers as opposed to the classical limits E J À EC

or E J ¿ EC ; both exhibit a certain uncertainty σ. For this reason, in the regime E J ≤ EC

which is relevant to this chapter, the junction cannot simply be represented by a particle in a

washboard potential 1 but rather by a wavefunctionΨ(φ), as illustrated in Fig. 6.2 [139]. This

purely quantum-mechanical property of φ and q also manifests itself in the experimental

I (VJ ) characteristics of the Josephson junction, in which it strongly alters the I (VJ ) curves

in comparison to the classical limits. For instance, the tunneling of the phase between two

adjacent minima in the washboard potential – similar to the tunnel effect discussed in Ch. 2,

Sec. 2.2.1 – can drastically reduce the maximum observable DC Josephson current [31, 40]. To

Figure 6.2 – (I) Classical representation of the junction phase as a particle in a washboardpotential, valid for E J À EC . (II) Quantum mechanical representation of the junction phase asthe wavefunctionΨ(φ) for the regime E J ≈ EC . Possible phase tunneling PT to the adjacentpotential minimum is indicated by the dashed arrow.

fully characterize the Josephson junction in our STM and clarify its properties it is necessary to

investigate this quantum-mechanical character of phase and charge. Although these quantum-

mechanical Josephson junctions attracted considerable experimental and theoretical efforts

in the years around 1990, questions on dissipation mechanisms and the transition from

sequential to coherent Cooper pair tunneling remained open, mainly due to the experimental

limitation that the Josephson coupling energy can not be easily tuned in experiments on

planar junction geometries [41, 140, 141, 142]. Moreover, varying the ratio of E J versus EC is

also of large interest in today’s science, especially in the context of Hamiltonian by design for

phase qubits [12]. In STM experiments the ratio E J versus EC can be readily tuned by varying

the tip-sample distance and for this reason, STM can serve as an ideal means for addressing

1This classical analogue represents the case E J À EC in which the phase is perfectly localized.

76

6.2. Defining the Quantum-Mechanical Regime

these fundamental problems of superconducting tunneling.

Within this chapter we investigate the quantum-mechanical character of the junction as well

as the energy dissipation of the junction phase φ(t ). To this end, we perform experiments on

current-biased and voltage-biased Josephson junctions, whereby this combined approach

provides total control of the parameter space. The analysis of the experimental data is partly

based on the P (E)-theory introduced in Ch. 4. The theoretical framework necessary for an-

alyzing I (VJ ) curves from current-biased experiments as well as φ will be presented in this

chapter.

6.2 Defining the Quantum-Mechanical Regime

Figure 6.1 shows that both the coherent as well as the sequential Cooper pair tunneling can be

observed when the capacitive charging energy EC is comparable to the Josephson coupling

energy E J (cf. Ch. 2, Sec. 2.5). However, the question still remains open for which range of EC

versus E J this manifestation of the quantum-mechanical nature of φ and q can be expected.

In other words, for which values of E J versus EC is the junction phase sufficiently localized in

a potential well so that the coherent tunneling of Cooper pairs can still be observed. We can

determine the boundaries of this regime when we investigate the extent of the wavefunction

Ψ(φ) in the washboard potential U (φ) as a function of E J versus EC . To this end we write again

the Hamiltonian of the Josephson junction, using λ= E J /(4EC ), (cf. Ch. 2, Eq. 2.35)2

H ′ = n2 −λcos(φ). (6.1)

This Hamiltonian is equivalent to the Hamiltonian of an electron which experiences a peri-

odic potential. In analogy to the Bloch functions for an electron in a crystal lattice (see e.g.

Ref. [143]), we can introduce a wavefunction

Ψ(φ) = eınφ. (6.2)

From the derivative ofΨ(φ) with respect to the phase,

∂

∂φΨ(φ) = ınΨ(φ), (6.3)

follows n =−ı∂/∂φ, which allows one to simplify the modified Hamiltonian in Eq. 6.1 to

H ′ =− ∂2

∂φ2 −λcos(φ). (6.4)

In this formulation, the Hamiltonian corresponds to the time-independent Schrödinger equa-

tion, which can be evaluated inφ space, H ′Ψ(φ) = εΨ(φ) and ε= E/(4/EC ). However, it cannot

be solved analytically due to the nonlinear cosine potential and thus it requires further means

2A similar approach for approximating this regime is given in Ref. [139].

77


of simplification. Assuming the phase to be sufficiently localized in the center of the potential

minimum we can expand the cosine term up to second order, giving

cos(φ) = 1− 1

2φ2 +O(φ3). (6.5)

This simplification yields a harmonic potential so that the resulting Schrödinger equation,

H ′Ψ(φ) = (− ∂2

∂φ2 + λ

2φ2)Ψ(φ) = εΨ(φ), (6.6)

corresponds to the Schrödinger equation of a harmonic oscillator [144]. This equation can

be solved analytically and its eigenfunctions, which represent the junction phase φ of the

Josephson junction in the phase space, are the Hermite functions

Ψn(x) = 1

π

√λ

2

1/4

1p2nn!

Hn(x)e−x2

2 , (6.7)

where x = 4√λ/2φ and Hn(x) denotes the Hermite polynomial [145]. The first two eigenfunc-

tions of the junction phase are shown in Fig. 6.3(a) for E J = EC inside the washboard potential.

The eigenenergies correspond to the harmonic oscillator eigenenergies,

εn =√

2λ(n +1/2),En =√8E J EC (n + 1

2), (6.8)

whereas the eigenenergy spectrum also reveals one of the major differences between the

quantum-mechanical and classical description; namely, that the junction phase does not “sit“

at the bottom of the potential as the classical particle does, but is elevated to a finite energy of

the lowest eigenstate, the zero point energy E0 =√

2E J EC .

Figure 6.3 – (a) Calculated first two eigenfunctions n = 0,1 of the junction phase for E J = EC .(b) Calculated n = 0 eigenfunction for different values of E J /EC , as indicated.

To determine the localization of the junction phase inside the potential, we investigate the

width of the eigenfunction in the phase representation. To this end, we calculate the standard

78

6.3. Theoretical Considerations

deviation σφ of the n = 0 eigenfunction, assuming that the junction is not in an excited state.

For this eigenfunction we find

Ψ0(x) = e−x2

2 → Ψ0(φ) = e

√λ8φ

2 = e−φ2

σ2φ → σφ = 4

√32EC

E J. (6.9)

A sufficient localization of the phase within one potential minimum requires the condition

σφ ≤ π. Hence, we are now able to define the lower limit of λ until which the phase is

sufficiently localized, giving

E J

EC≥ 32

π4 . (6.10)

The standard deviation for the junction charge directly follows from the commutator relation

between charge and phase (cf. Ch. 2, Sec. 2.5), giving σφ = 4√

E J /32EC . Figure 6.3(b) displays

the calculated n = 0 eigenfunction inside the washboard potential for these two values of

E J versus EC which define the upper and lower boundaries of the range in which quantum-

mechanical phenomena should effect the properties of the Josephson junction.

If we put in a typical capacitance of the tunnel junction in our STM, C J ≈ 2.5fF (cf. Ch. 4), we

find a limiting value to smaller numbers for the Josephson coupling energy of E J ≈ 16µeV.

Comparing this value with the values given in Fig. 4.1 in Ch. 4, we find that the phase should be

sufficiently localized inside a potential minimum for tunneling conductance values exceeding

GN /G0 > 0.1. This value is well within the tunneling conductance range accessible to our

experiment. From the same figure we also find that the upper limit of the quantum-mechanical

regime, E J /EC ≈ 3, is well above the quantum of conductance, GN /G0 = 1, and exceeds the

maximum value found in experiments E J /EC ≈ 1.63.

Nevertheless, the quantum-mechanical regime is clearly accessible for our experiment and

we should be able to observe footprints of this regime in the experimental I (VJ ) curves for

GN /G0 ≥ 0.1. As presented in Ch. 4, Sec. 4.5 we could already observe first indications of this

quantum-mechanical regime when analyzing the voltage-biased I (VJ ) curves in the limit of

GN ≈G0.

6.3 Theoretical Considerations

Within the last section, we showed, in a simple approximation, that for a particular range of

the tunneling conductance accessible to our experiment the Josephson junction will be in a

quantum-mechanical state in which neither its phase or charge are perfectly defined. To inves-

tigate this state we perform voltage-biased and current-biased experiments on the Josephson

junction. The theoretical framework for analyzing the sequential tunneling characteristics of

voltage-biased experiments has already been presented in Ch. 4. This section will focus on

presenting theory, which describes the bound state of the junction in a washboard potential

minimum at a given junction phase φ(t ) as well as the temporal evolution φ(t ). In particular,

79


the theory focuses on two experimental quantities, which are the switching current IS and the

retrapping current IR , as they were introduced in Ch. 2, Sec. 2.3 [59, 60, 146]. These quantities

provide direct access to the quantum behavior of the phase and the dissipative interaction

of the junction phase with the environment and, additionally, they are easy to determine

experimentally.

6.3.1 Retrapping into the Zero Voltage State

Figure 6.1(b) shows that the I (VJ ) curves of the Josephson junction in our STM are strongly

hysteretic. Such characteristics are typical for junctions in the limit E J À EC whose phase

dynamics can be treated classically, but it is not a priori to be expected from junctions in

which E J is comparable to EC . Despite this, in the last section, we could approximate a range

of E J versus EC values, in which the phase is still sufficiently localized in the washboard

potential minima and we found that this range of tunneling resistance values is accessible

to our experiment. Therefore, we present theory for analyzing the retrapping characteristics

of the Josephson junction that treats the phase as a classical variable – an approach that is

sufficient to explain our experimental observation as the later analysis will reveal (cf. Sec. 6.7).

Starting from the very first formulation of the Josephson effect, it is possible to derive a

differential equation for the temporal evolution of the junction phase, as was introduced

in Ch. 2, Sec. 2.3. This equation, which can be solved by numerical means, facilitates the

investigation of the temporal evolution of the phase φ(t) as the movement of the phase,

represented by a particle, through the washboard potential. While the external current bias IB

tilts the washboard potential and serves as a driving force, the particle also experiences friction

during its movement through the potential landscape. The strength of this friction depends on

various parameters, such as the junction capacitance C J or the tunneling resistance RN [59, 60]

and strongly influences the dynamics of φ(t ). Considering the I (VJ ) curve of a current-biased

measurement displayed in Fig. 4.1(b), for instance, we clearly observe a strong hysteresis in

the tunneling characteristics and that the retrapping into the zero voltage state occurs close to

zero current bias. According to the theoretical introduction (cf. Ch. 2, Sec. 2.3), the existence of

a hysteresis loop indicates underdamped junction dynamics, Q = RDC C J ×√

2eI0/(C Jħ) À 1.

Inserting typical values for our experiment in the tunnel regime (I0 ≈ 5nA, RN ≈ 250kΩ and

C J ≈ 2.5fF, as given in Ch. 4) we indeed find Q ≈ 30 and we can conclude that, for standard

STM experiments, the junction dynamics are highly underdamped 3.

The retrapping into the zero voltage state occurs when the driving energy from the current

bias cancels out with the energy dissipated due to friction and we again write the condition

3In principle, underdamped dynamics are common to small area Josephson tunnel junctions, since the ca-pacitance of the contact strongly decreases with the junction area. However, it is possible to create overdampeddynamics by means of advanced circuitry. Here, the junction dynamics are effectively decoupled from the environ-ment and dissipative elements, such as resistors, are introduced to enhance the damping. This approach is inparticular interesting in order to reduce phase diffusion in small area Josephson junctions and very descriptiveexamples of this approach can be found in Refs. [61, 62].

80


for retrapping [59, 60]:∫ π

−πdφ

IR

I0= 1

ω20τ

∫ π

−πdφφ(t ). (6.11)

For a junction that is shunted with an ohmic resistor this equation can be readily solved and

yields the simplest form of the retrapping current [59],

IR = 4

π

I0

Q, (6.12)

which clearly displays the strong dependence of the retrapping current on the energy dissipa-

tion of the junction phase, as expressed by the quality factor Q. However, many experiments on

Josephson tunnel junctions do not comprise an ohmic shunt but instead a voltage-dependent

quasiparticle resistance shunt RQP(VJ ). Here, the I (VJ ) characteristics deviate from the ohmic

behavior for VJ ≤ 2∆ and the current is carried by quasiparticle excitations, as is shown in

Fig. 6.4 (cf. Ch. 2, Sec. 2.1). Given an experiment at zero temperature and a perfect super-

conducting BCS gap, the differential resistance ∆VJ /∆IQP will diverge for voltages smaller

than twice the superconducting gap, VJ ≤ 2∆. Accordingly, the quasiparticle resistance RQP

will strongly exceed the ohmic resistance RDC. As a consequence, the dissipation into the

environment is inhibited and the retrapping of the junction into the bound state occurs at

much smaller values of the current bias in comparison to an ohmic shunt [146].

Figure 6.4 – Comparison of a calculated ohmic current IDC (VJ ) and a calculated quasiparticlecurrent IQP(VJ ) at a tunneling resistance of RT = 109kΩ.

Calculating the retrapping current for a tunnel junction, therefore, requires one to consider

all possible dissipative channels that constitute an effective in-gap resistance Reff. Such

a calculation was first performed by Chen, Fisher and Leggett (CFL) [147]. The authors

incorporated all possible voltage-dependent quasiparticle and pair transfer processes, as

given in Ref. [56], into a quantum-Langevin equation that describes the dissipative time

81


evolution of the phase [148, 147]. Assuming zero temperature, the authors employed Eq. 6.11

to calculate an expression for the retrapping current, giving [147]

IR,T=0 = 8∆2

eħω0RNe−

2π∆ħω0 . (6.13)

This formulation of the retrapping current represents its ultimate limit that can be reached if

dissipative in-gap quasiparticle currents do not exist. Hence, the particle moving down the

washboard potential can only dissipate energy via breaking up Cooper pairs at VJ = 2∆ and

the junction will therefore retrap into the potential minimum at smallest current bias values.

For the same reason, it is experimentally very challenging to reach this minimum retrapping

current as we will also see in the later analysis (cf. Sec. 6.7). In most experiments, lifetime

effects of Cooper pairs, Andreev reflections or finite temperatures induce in-gap quasiparticle

currents at VJ < 2∆ that increase dissipation and, thereby, increase the current at which the

junction retraps.

In an initial experimental work, Kirtley et al. addressed the influence of thermally excited

quasiparticles on the retrapping current [140]. The starting point of their analysis, which is

based on the CFL model, is an effective in-gap resistance for VJ < 2∆

1

Reff= 1

RQP+ 1

RQPPcos(φ), (6.14)

which includes the quasiparticle resistance RQP as well as the quasiparticle-pair interference

resistance RQPP. Employing Eq. 6.11 the authors first simplified the equation for the retrapping

current to

I CFLR = 4I0

πω0C J

1

Reff= 4I0

πω0C J

[1

RQP+ 1

3RQPP

]. (6.15)

Secondly, assuming a Boltzmann-type reduction of the quasiparticle resistance inside the gap,

1/RQP ∝ exp(−∆/kB T ) and the temperature such that ω0 ¿ kB T ¿∆ – this energy window

corresponds typically to a temperature of a few Kelvin – the authors derived the following

expression for the retrapping current

I CFLR = 8I0

3Q

∆

kB Te−∆/kB T . (6.16)

The two presented, yet quite different, examples for the relation between the dissipation and

the retrapping current already illustrate that the energy dissipation strongly depends on the

type and strength of the in-gap quasiparticle currents. To obtain a proper description for our

experiment, we compare the thermal energy and the energy of the Josephson plasma mode

ħω0 with values from Ch. 4 in Fig. 6.5(a). It is apparent that, for our experimental conditions,

the thermal energy, kB T , is much smaller than the energy of the plasma mode for all values of

82


the tunneling conductance, for which reason Kirtley’s model for calculating IR is not applicable

in our case.

Moreover, a typical experimental quasiparticle excitation spectrum, as shown in Fig. 6.5(b),

reveals broadened quasiparticle excitation peaks as well as a finite conductivity inside the

superconducting gap, i.e. the presence of dissipative in-gap quasiparticle currents. In our case,

both phenomena result from the lifetime effects of Cooper pairs discussed in Ch. 2, Sec. 2.2.4.

We also find that both the sample gap, ∆1 ≈ 800µeV, and the tip gap, ∆2 ≈ 300µeV, are much

larger than kB T and ħω0. Although in our experiment the influence of the temperature is

negligible, the existence of dissipative in-gap currents will make it impossible to employ the

zero temperature limit of the CFL theory (cf. Eq. 6.13) in order to descibe the retrapping

current in our experiment.

Figure 6.5 – (a) Comparison of Josephson plasma energy (values taken from Ch. 4) and thermalenergy as a function of the normalized tunneling conductance. (b) Experimental quasiparticleexcitation spectrum measured at GN = 0.11G0 and T = 15mK.

Hence, we have to derive an appropriate description of the sub-gap dissipation that accounts

for the in-gap quasiparticle currents induced by lifetime effects of Cooper pairs. To this end, we

start with the Dynes equation in order to model the dissipative quasiparticle current through

our tunnel junction according to

IQP(VJ ) = 1

eRN

∫ VJ

0ρt (E −V )ρs(E)dE , (6.17)

ρi (E) = ℜ

E − ıΓi√(E − ıΓi )2 −∆2

i

(6.18)

(cf. Ch. 2, Sec. 2.2.4). Here, ρt and ρs denote the superconducting density of states for tip and

sample, respectively. We are interested in the in-gap quasiparticle resistance RQP =VJ /IQP for

VJ ¿∆1 +∆2 which allows us to simplify Eq. 6.17 further on. To this end, we expand Eq. 6.18

83


around E = 0 using the relations ε= E/∆ and γ= Γ/∆:

ρi (E) = γi√1+γ2

i

+ 3ε2i γi

2(1+γ2i )5/2

+0(ε3i ). (6.19)

If we consider only the zero order term in εwe can directly evaluate the integral in Eq. 6.17. This

step yields the expression for the quasiparticle tunneling current inside the superconducting

gap ,VJ ¿∆1 +∆2,

IQP(VJ ) ≈ VJ

RN

γ1γ2√(1+γ2

1)(1+γ22)

, (6.20)

from which we readily obtain the quasiparticle resistance:

1

RQP= γ1γ2√

(1+γ21)(1+γ2

2)

1

RN. (6.21)

Using Eq. 6.15 for the retrapping current, as derived by Kirtley et al. [140], we can calcu-

late the retrapping current in the zero temperature approximation for a lifetime broadened

superconducting density of states using

IR = 4

π

I0

Q

γ1γ2√(1+γ2

1)(1+γ22)

. (6.22)

For the derivation of this expression we neglected the RQPP term in Eq. 6.15 for the simple

reason that we estimate the gap filling due to Cooper pair lifetime effects to be relatively strong

in comparison to gap filling effects by Andreev reflections in the tunneling regime GN ¿G0.

6.3.2 Stability of the Zero Voltage State

The previous section treated the time evolution of the phase φ 6= 0 in a classical manner

by investigating the dissipative motion of the phase through the washboard potential. The

classical treatment was possible since the localization of the phase – represented by the

wavefunctionΨ0(φ) = exp(−φ2

σ2φ

) – inside the potential minimum was assumed to be sufficiently

large. However, Ψ0(φ) is still of finite width σφ and, therefore, tunneling processes to the

adjacent minimum are in principle also possible, as illustrated in Fig. 6.6 (cf. Ch. 2, Sec. 2.2.1).

On the one hand, this tunneling strongly reduces the amplitude of the DC Josephson current

to values IS ¿ I0. On the other hand, for small bias currents I ≤ IS tunneling to adjacent

minima results in a non-average φ(t ), such that the DC Josephson effects occurs at VJ 6= 0. In

this situation, the DC Josephson current exhibits a finite slope R0 =VJ /I for I ≤ IS , as is also

apparent in the exemplary I (VJ ) curve shown in Fig. 6.1(b) [31, 41, 139]. Hence, in comparison

to the dynamics of φ(t ), the stability of the zero voltage state at constant phase is much more

effected by the quantum-mechanical nature ofφ(t ) and q . For this reason, this section focuses

84


on a quantum-mechanical description of the zero voltage state as well as the influence of

quantum mechanics on the measurement quantities.

The finite slope in the coherent Cooper pair tunneling current can be explained on the basis

of Fig. 6.6(a): If the washboard potential is slightly tilted by an applied bias current the barrier

height U in the downhill direction is reduced, U→ = 2E J −ħπI0/2e, whereas it is increased in

the uphill direction, U← = 2E J+ħπI0/2e [139]. Hence, the probability for tunneling downwards−→PT is larger than for tunneling upwards

←−PT , which results in a net rate of tunneling with a

probability ∆P = −→PT −←−

PT > 0. Provided that damping of the junction is large enough, the

junction phase will relax down into the lowest energy level of the adjacent minimum and

remain in the bound state. Still, these phase slips correspond to a small, albeit non-zero, φ(t ),

such that a small voltage drop VJ occurs across the junction. Assuming a sufficient localization

of the phase inside the potential and a bias current I ¿ I0, the resistance R0 resulting from

these quantum-mechanical phase slips can be evaluated [40, 108, 139]

R0 = VJ

I= ħ

2e

1

I0

dφ

d t≈ ħ

2e

1

I0(←−PT −−→

PT ), (6.23)

−→PT = ω0

2πχ

4

√U

ħω0e−

√U

ħω0s. (6.24)

Here, χ= 52.1 and s = 7.2 are numerical constants, as given in Ref. [139]. The state of finite

phase slips is only stable as long as the junction phase experiences sufficient dissipation. At

some critical current IS the dissipation is too small and the phase will start moving down

the potential wall at a velocity φ(t). That situation corresponds to the switching out of the

zero voltage state, which can be observed experimentally in the tunneling characteristics.

Although it is difficult to introduce continuous energy dissipation into a quantum- mechanical

framework, Iansiti et al. succeeded in, at least, making an estimate for the current at which

the junction leaves its bound state. To this end, they assumed that the work performed by the

current to induce a phase shift of π – corresponding to a translation to the adjacent minimum

– must be smaller than the binding energy of the lowest eigenmode inside the potential. In this

way, the authors derived the switching current as [139]4

IS = e

4ħE 2

J

EC. (6.25)

Interestingly, this estimate predicts a quadratic dependence of the switching current on the

Josephson coupling energy for the regime E J ≈ EC . In fact, Fig. 6.1(b) shows us that the

switching current is of the same value as the maximum Cooper pair current peak found

in voltage-biased measurements. For these experiments, we could show in Ch. 4 that the

amplitude of the Cooper pair current scales with I ∝ E 2J .

It should be noted that the effects of phase tunneling on the junction properties were presented

4In a similar approach Ferrell et al. obtained the same result [149].

85


Figure 6.6 – (a) Illustration of the possible tunneling processes for the wavefunctionΨ(φ). (a)Energy bands spectrum of the junction in the charge space for λ= 0.4, adopted from Ref. [150].The interband transition PZ causing phase slips and premature switching out of the zerovoltage state is indicated.

in phase space. On the one hand, this approach facilitates a very descriptive explanation of the

physical phenomena, also referring to the particle in a washboard picture. On the other hand,

for our experiment we find that, for most of the tunneling resistance range, the Josephson

coupling energy is a bit smaller than the charging energy, λ= E J /EC ≤ 1. For this reason, the

phase will experience a certain amount of delocalization which makes a precise treatment of

the discussed phenomena challenging. Although the presented derivations on R0 and IS are

at least qualitatively correct, for quantitative conclusions it appears favorable to move to a

description in the charge space [139, 150, 151, 152, 153].

Here, the energy level diagram for the phase in the washboard potential is replaced by a

“band structure“ E(q) for the charge on the junction, which is similar to the band structure

of an electron in a crystal. For example, Fig. 6.6(b), displays such a band structure for λ= 0.4.

The bandwidth scales with the charging energy EC , whereas the band gap at the edge of the

Brioullin zone, at q = e, is on the order of the Josephson coupling energy. In this picture, an

intraband transition in the lowest band to the next Brioullin zone corresponds to the transfer

of one Cooper pair across the tunnel junction. The discussed phase slips for I ¿ IS in phase

space here correspond to interband transitions into a higher band, as indicated in Fig. 6.6(b).

This process is in analogy to Zener tunneling as observed in pn-diodes, for instance (see e.g.

Ref. [143]). If the bias current increases, the probability for Zener tunneling also increases. At a

critical current IS the junction is not able to discharge the Cooper pairs from the capacitor

anymore and the Zener breakdown occurs, i.e. the junction switches out of the zero-voltage

state [150, 40, 139], as in analogous to the Zener breakdown in diodes.

This explanation might appear less descriptive than the presented explanation in phase space,

yet the charge space representation facilitates a quantitative determination of the switching

current in the limit E J ≤ EC . Based on the theoretical framework for Zener tunneling, Iansiti et

86

6.4. Numerical Simulation of the Phase Dynamics

al. could derive an expression for the switching current, giving [40]

IS,Z = πe

8ħE 2

J

EC. (6.26)

In comparison to the expression for IS derived in phase space we find that IS,Z is only larger

by a factor of π/2 but shares the same quadratic dependence on E J .

Finally, it should be noted that all the presented theoretical considerations have neglected the

influence of thermal fluctuations described in Ch. 2, Sec. 2.4. The main reason is that for all

values of the tunneling conductance, we find that, in our case, the thermal energy is always

(much) smaller than the Josephson coupling energy (cf. compare Fig. 4.1). Hence, thermal

activation of the phase over the washboard potential barrier is unlikely to occur and thus

tunneling through the barrier will be the predominant source of premature switching out of

the zero voltage state.

6.4 Numerical Simulation of the Phase Dynamics

Investigation of the phase dynamics φ by means of tunnel experiments can be complemented

by performing numerical simulations on the temporal evolution of φ(t) as initially demon-

strated in Ref. [146]. To this end, we start out with the classical differential equation for φ(t)

with respect to time, as introduced in Ch. 2, Sec. 2.3, which we write again here [55, 59, 60, 146]:

I = I0 sin(φ(t ))+ IQP(h

2eφ(t ))+C

ħ2eφ(t ). (6.27)

This equation can be solved by numerical means using a differential equation solver such

as the ode45-solver in Matlab. For the later analysis, we limit ourselves to the simulation of

the retrapping characteristics of the Josephson junction, since the switching characteristics

demand a quantum-mechanical treatment. For this reason, the boundary condition φ(t =0) 6= 0 has to be chosen. In order to calculate a full I (VJ ) tunneling spectrum for a range of

current bias values I = I0, I2, . . . , In it is necessary to solve this equation for each individual

value of the current bias In , which itself serves as an input parameter. From the solution φ(t )

we directly obtain its derivative φ(t ), which evolves in time as shown, for example, in Fig. 6.7.

As soon as φ(t ) converges to a constant value, the voltage drop across the tunnel junction can

be calculated using VJ = (ħ/2e)φ.

We model the dissipative quasiparticle current IQP( h2e φ(t)) via a lifetime broadened BCS

density of states as given in Eq. 6.17 and 6.18 (cf. Ch. 2, Sec. 2.2.4).

Fig. 6.8 displays simulated I (VJ ) tunneling curves focused on the in-gap region VJ ¿∆1 +∆2

for all individual values of the tunneling resistance RN as given in Sec. 6.5. For the simulation

we used C J = 2.53fF, ∆1 = 800±20µeV and ∆1 = 309±20µeV (cf. Sec. 6.6) as well as lifetime

broadening parameters Γ1 = 20µeV and Γ2 = 63µeV (cf. Sec. 6.7). By visual inspection of the

87


Figure 6.7 –φ(t ) and φ(t ), as obtained by numerically solving Eq. 6.27 for a chosen bias currentof I = 100pA.

I (VJ ) curves, the increase in the retrapping current for decreasing tunneling resistance values

is apparent. Moreover, the in-gap quasiparticle current is enhanced for reduced values of RN

which descriptively illustrates its significance as a dissipation channel effecting IR .

Figure 6.8 – I (VJ ) tunneling curves as calculated from the simulated φ values plotted fordifferent values of RN , as indicated.

6.5 Comparison of Current-biased and Voltage-biased Experiments

To investigate the quantum-mechanical properties of the Josephson junction in our STM

we performed experiments in which we applied both a bias voltage and bias current to the

88

6.5. Comparison of Current-biased and Voltage-biased Experiments

Josephson junction 5. In this way, we can fully control the parameter space of the Josephson

junction: By analyzing the I (VJ ) experiments we can precisely extract the Josephson critical

current I0 as well as the capacitance of the tunnel junction C J . In conjunction with the

tunneling resistance RN these parameters serve as valuable input parameters for analyzing

the VJ (I ) curves 6. These measurements were performed as described in Ch. 3, Sec. 3.4 and 3.5,

Figure 6.9 – (a-c) Comparison of voltage- and current-biased experiments for different valuesof RN as indicated. (d) Exemplary I (VJ ) curve in which the relevant points for extracting IS andIR are indicated. Moreover, the point of minimum quasiparticle current, Vmin, is indicated.

using a large range of tunneling resistance values from RN ≈ 50 kΩ to RN ≈ 300 kΩ. Figure 6.9(a-

c) displays the experimental I (VJ ) curves for three different values of RN and also compares

the results obtained by voltage- and current-biased experiments. As it was already shown in

Fig. 6.1, the I (VJ ) curves of both bias techniques perfectly match and only deviate for regions

of negative differential conductance, i.e. d I /dVJ < 0. In particular, the agreement of both

curves in the in-gap quasiparticle current for VJ < 1mV and for the range of coherent Cooper

pair tunneling around VJ = 0 will be of importance for the later analysis.

We assign the tiny deviations found between the two I (VJ ) curves to small deviations in the

tunneling resistance of both experiments. Fig. 6.10 compares the fitted RN values and their

5It should be noted that the inconsistency in terminology by using both the terms tunneling conductance andtunneling resistance is on purpose and reflects the common practice in the the respective contexts.

6For the sake of simplicity the tunneling characteristics of both current- and voltage-biased experiments will belabeled I (VJ ) in the following, whereas the bias source will be mentioned where necessary.

89


Figure 6.10 – Comparison of fitted tunneling resistance values RN of voltage-biased andcurrent-biased experiments as well as their relative deviation ∆RN .

relative deviation∆RN (cf. Ch. 3, Sec. 3.4). We find that, for most measurements, the deviations

are small ∆RN ≤ 5% and might originate from a systematic error in adjusting the tunneling

resistance by manually adjusting the tip-sample distance with the z-piezo. We find a larger

deviation only for one value at RN = 148kΩ. This could be induced, for instance, by a surface

adsorbate in the tunnel contact leaving the contact instable. Nevertheless, the good agreement

between the RN values of voltage- and current-biased experiments allows us to combine results

from both measurement techniques for investigating the quantum-mechanical properties of

the Josephson junction in our STM.

From the I (VJ ) curve, as for example shown in Fig. 6.9(d), we extract the retrapping current

IR as the value of the bias current at which the voltage across the tunnel junction VJ rapidly

drops to zero voltage, when the bias current is continuously decreased starting from IB > I0.

We extract the switching current IS as the value of the bias current at which the voltage across

the tunnel junction VJ rapidly increases from zero to voltages on the order of ∆when starting

from IB = 0A.

6.6 Analysis of the Voltage-biased Experiments

In this section we analyze the I (VJ ) curves obtained from voltage-biased experiments on the

Josephson junction in our STM for all values of the tunneling resistance shown in Fig. 6.10.

We employ P (E)-theory and fit the I (VJ ) curves in the same fashion demonstrated in Ch. 4,

Sec. 4.4. Figure 6.11 displays the experimental I (VJ ) characteristics and the corresponding

fit, which is split up into the Cooper pair tunneling current, modeled by P (E)-theory, and

90

6.6. Analysis of the Voltage-biased Experiments

the quasiparticle background. This separation will be of particular importance in the later

analysis of the retrapping current. From Fig. 6.11 we see that the fit nicely models each single

contribution to the total Cooper pair current. From such a fit we readily extract the Josephson

coupling energy E J , the junction capacitance C J and the cubic background for each single

value of the tunneling resistance 7.

Figure 6.11 – I (VJ ) tunneling curves and the P (E )-fit to data as well as the in-gap quasiparticlecurrent, as modeled by the parameter c2.

In Figure 6.12(a) we plot the experimental Josephson critical current I0, that we calculated

from the fitted E J values as a function of the normalized tunneling conductance GN /G0.

Furthermore, we compare these values with theoretical values that we calculated using the

Ambegaokar-Baratoff formula (Ch. 2, Eq. 2.21) [8]. As already demonstrated in Ch. 4, we can

extract the gap values for the two superconducting electrodes from the d I /dVJ -spectrum

and its derivative shown in Fig. 6.12(b). In this case, the superconducting properties of the

tip were such that both spectra only feature the quasiparticle excitation peak at ∆1 +∆2 =1109±10µeV. We therefore estimate the sample gap value to ∆1 = 800±20µeV – a value that

represents the average value of the sample gaps found in our experiments – and yield a tip gap

∆2 = 309±10µeV.

The fitted and calculated critical current values, shown in Fig. 6.12(a), are in good agree-

ment along the entire range of tunneling conductance values, as we also found it for the

experiment discussed in Ch. 4, Sec. 4.6. For this set of data, the deviations are slightly larger,

especially for GN ≥ 0.1G0. However, on average, we find an acceptable standard deviation

between theory and experiment of σ∆I0 = 9%. Moreover, we find an unexpected behavior

for I0 at a conductance of about 0.09G0, which corresponds to a tunneling resistance of

RN = 1/(0.09G0) = 143kΩ). We already observed this abnormality when comparing RN values

7To minimize uncertainties in the fit, we initially determined the lowest eigenmode of the tip ν0 and its qualityα to ν0 = 42.1GHz and α= 0.71, respectively, and afterwards kept these fixed when fitting the dependence of theI (VJ ) curves on RN (cf. Ch. 4, Sec. 4.5.)

91


Figure 6.12 – (a) Comparison of experimental and theoretical I0 values and their relativedeviation ∆I0 plotted as a function of RN . (b) First d I /dVJ and second d 2I /dV 2

J derivative ofthe quasiparticle tunneling current, which facilitate determination of the gap values ∆1 and∆2.

from voltage- and current-biased experiments in the last section.

In Fig. 6.13(a), we plot the fitted junction capacitance C J as a function of RN . Just as we found

from the analysis of voltage-biased experiments in Ch. 4, C J does not depend on RN . Hence, we

can calculate an average value of C J = 2.53±0.11fF, which we will use for the following analysis.

In contrast, the fit parameter c2, which models the in-gap quasiparticle current, exhibits a

strong dependence on RN . Whereas c2 remains almost constant for larger values of RN , it

drastically increases when RN is reduced below 100 kΩ. The constant contribution originates

from the lifetime-broadened superconducting density of states (cf. Ch. 2, Sec. 2.2.4). We can fit

the strong increase in c2 with a power law function c2 = A/RξN as shown in Fig. 6.13(b). From

the fit we find ξ=−1.94±0.07, A = (1.02±0.82)×109 1/(AV). The square dependence of the

in-gap current on the tunneling resistance is a strong indication that the dominant source

92

6.6. Analysis of the Voltage-biased Experiments

Figure 6.13 – (a) Junction capacitance C J as extracted from the P (E)-fits plotted as a functionof RN . (b) Cubic background parameter c2 as extracted from the P (E)-fits as well as the fit tothe c2 parameter plotted as a function of RN .

of the gap filling is Andreev reflections, a typical feature in transparent tunnel contacts, as

introduced in Ch. 2, Sec. 2.2.3. At this point we keep the analysis of the in-gap quasiparticle

current on the presented phenomenological level, which should be sufficient for the later

investigation of the phase dynamics φ(t ). A more detailed analysis of this quasiparticle current

is certainly desirable, also in view of the multiple Andreev reflection processes, but will be part

of future work.

In Sec. 6.3.1 we could derive a relation between the phenomenological lifetime parameter

Γ and the retrapping current of the Josephson junction. To determine the Γ values of the

superconducting gaps, it is necessary to investigate the quasiparticle excitation spectrum

(cf. Ch. 2, Sec. 2.2.4). To this end, we calculate the first derivative d I /dVJ of the quasiparticle

tunneling current from the experimental I (VJ ) curves, as shown in Fig. 6.14(a). We perform

93


Figure 6.14 – (a) d I /dVJ spectrum normalized to the normal metal density of states ρN

measured at RN = 86kΩ and the fit to data using the extended Maki model. (b) Lifetimebroadening parameter of the sample Γ1 and the tip Γ2, as extracted from the Maki fits, plottedas a function of RN .

this calculation for all values of the tunneling resistance RN given in Sec. 6.5. It should be noted

that the large feature around zero voltage in the d I (VJ )/dVJ curves of Fig. 6.14(a) is an artifact

that results from differentiating the current sigal. We can fit the resulting d I /dVJ spectra

using an extended Maki model [154, 155, 156]. In comparison to the BCS model introduced

in Ch. 2, the Maki model used for this study contains an additional depairing parameter ζ

that effectively reduces the amplitude of the coherence peaks at VJ =∆1 +∆2. The modified

superconducting density of states reads as

ρS,M aki (E) = ρN

2sgn(E) Re

u±√u2±−1

. (6.28)

94

6.7. Analysis of the Retrapping Current

Here, u+ and u− are implicitly defined functions:

u± = E − ıΓ

∆+ ζu±√

1−u2±

. (6.29)

Fig. 6.14 shows a fit to an experimental d I /dVJ spectrum using this model. From the fits,

we extract the lifetime parameter Γ1 and Γ2, which we plot as a function of the tunneling

resistance RN in Fig. 6.14(b). Whereas Γ1 of the sample does not show any dependence on

RN , Γ2 of the tip slightly decreases for smaller RN values. However, this behavior is counter

intuitive since we actually expect the gap to fill up when RN is reduced, which should result

in a strong increase of Γ2. A possible explanation is that the dependence of the coherence

peaks on the Γ-value is much stronger in comparison to the gap-filling introduced by this

parameter. In comparison, the c2 parameter, which includes contributions from both Γ and

Andreev reflections and also models the gap-filling, does strongly depend on RN . Nevertheless,

for the later analysis of the in-gap quasiparticle resistance we can assume Γ to be independent

of RN and employ average values for the lifetime parameter of Γ1 = (1.40± 1.28)µeV and

Γ2 = (25.3±3.8)µeV, respectively.

6.7 Analysis of the Retrapping Current

In Section 6.3.1 we introduced the concept that the retrapping current IR serves as a probe

for the dissipative interaction of the Josephson junction with its environment and that for

tunnel junctions this dissipation originates from breaking up Cooper pairs. To analyze this

dissipative interaction we extract the retrapping current IR as defined in Sec. 6.5 and plot it as

a function of the tunneling resistance RN in Fig. 6.15. We find that IR is strongly dependent

on RN and increases by two orders of magnitude when RN is reduced by only half an order of

magnitude. In view of Eq. 6.15 this observation signifies that the effective in-gap resistance

Reff drastically changes on this tunneling resistance interval.

To investigate the dependence IR (RN ), we employ Eq. 6.22 to fit the retrapping current, which

is displayed in Fig. 6.15. We find that the fit nicely models the retrapping current for larger

values of the tunneling resistance, RN ≥ 150kΩ but underestimates its value when the junction

becomes more transparent, RN ≤ 150kΩ. For fitting, we used the average values for C J , ∆1 and

∆2 as given in Sec. 6.6. For the fit, we keep one lifetime parameter constant at a reasonable

value, Γ1 = 20µeV and for the second lifetime parameter we find Γ2 = (63±4)µeV. Despite

the fact Γ1 was deliberately kept constant, we find that both Γ values obtained from fitting

the retrapping current are (much) larger than the Γ values found from the Maki fits in the

last section. On the one hand and as discussed before, the Γ parameter in the extended Maki

model might be relatively insensitive to the very small in-gap density of states (cf. Fig. 6.14(a))

in comparison to its drastic influence on the coherence peak width and amplitude [54]. For

this reason, the in-gap density of states, which strongly affects the retrapping current via the

effective in-gap resistance, might not be perfectly modeled by the Maki fits. On the other hand,

95


Figure 6.15 – Experimental and simulated retrapping current and the fit to experimental dataplotted as a function of RN . For the simulated retrapping current the error bars are includedin the symbols.

the quantitative modeling of IR on an analytical basis is difficult to achieve [148, 147] and, so

far, published studies on this topic only obtained a qualitative agreement between experiment

and theory [140].

The deviation between fit and experimental data for RN ≤ 150kΩ can be better understood if

we also consider the in-gap quasiparticle currents observed in voltage-biased experiments

in Sec. 6.6. Here, the in-gap current was nicely modeled by the c2 parameter. We found that

this parameter, shown in Fig. 6.13, strongly increases for RN ≤ 150kΩ and we could assign this

increase to the likely onset of Andreev reflections causing the observed in-gap quasiparticle

currents. In contrast, for larger values of the tunneling resistance, c2 is nearly constant and

displays the finite in-gap current that results from lifetime effects, as expressed by Γ. Hence, for

RN ≥ 150kΩ the effective in-gap resistance Reff is significantly reduced by the lifetime effects

of Cooper pairs, whereas for smaller values of the tunneling resistance the in-gap resistance is

dominated by Andreev reflections. In comparison, both Γ-parameters that model the effective

in-gap resistance in Eq. 6.22 do not comprise an implicit RN dependence. For this reason, Γ

cannot account for a reduction in Reff due to Andreev reflections and the analytic expression

given in Eq. 6.22, which relates lifetime effects of Cooper pairs to the retrapping current, is not

applicable in this regime.

We obtain very similar results for the retrapping currents that we extract from the simulated

I (VJ ) curves introduced in Sec. 6.4 8. For the simulation we used the Γ values that we obtained

from the fit to the experimental retrapping currents using Eq. 6.22. In Fig. 6.15 we plot the

simulated retrapping currents as a function of the tunneling resistance. Remarkably, the

IR values found from the simulations almost perfectly reproduce the fitted IR curve. As a

8The IR values are determined in the same fashion as demonstrated in Sec. 6.5.

96

6.7. Analysis of the Retrapping Current

consequence, the simulated IR values match their experimental counterparts for RN ≥ 150kΩ

but also underestimate their values for RN ≤ 150kΩ. On the one hand, this is certainly

expected, since the dissipative quasiparticle current in the simulation is of the same form as

the current used for the derivation of Eq. 6.22. On the other hand, this result corroborates the

validity of our assumptions made for deriving Eq. 6.22. Hence, Eq. 6.22 can be regarded as a

proper description for modeling the retrapping currents in the regime RN ≥ 150kΩ.

Nevertheless, we can address the enhanced in-gap quasiparticle current arising from Andreev

reflections on a phenomenological level in order to correctly model the experimentally ob-

served retrapping currents. To this end, we again look at the c2 parameter from the P (E)-fits

in Sec. 6.6. Comparing voltage- and current-biased experiments in Fig. 6.9(a-d) we could show

that the in-gap quasiparticle currents from both measurements are identical. Therefore, the

effective in-gap resistance at which the junction retraps can be calculated on the basis of the

c2 parameter. In detail, we calculate the differential conductance d I /dVJ of the quasiparticle

current according to d I /dVQP = 3c2 ×V 2. Figure 6.9(a-d) shows that the retrapping occurs

at the same voltage of Vmin ≈ 330µeV independent of the tunneling resistance. Hence, the

effective in-gap resistance at which the junction retraps calculates as

Reff =∣∣∣∣ 1

d I /dVJ

∣∣∣∣V =Vmin

= 1

3 c2V −2

min. (6.30)

From these values for the effective in-gap resistance, we can easily calculate the retrapping

current from Eq. 6.15. In Fig. 6.16 we plot these IR values together with the experimentally

determined IR values as a function of RN . We find quantitative and qualitative agreement

between calculation and experiment over the entire range of tunneling resistance values. In

particular, in the regime of RN ≤ 150kΩ, where Andreev reflections strongly reduce Reff, the

calculation now correctly models the resulting strong increase in IR .

Figure 6.16 – Comparison between experimental retrapping values and and calculated valuesusing the c2 parameter plotted as a function of RN .

97


Together, our experimental results demonstrate that the dependence of the retrapping current

on the tunneling resistance can be separated into two regimes. These regimes are determined

by two different dissipative interactions of the junction phase with the environment. We find

that for opaque tunnel junctions – in our case RN ≥ 150kΩ – dissipation is dominated by life-

time effects of Cooper pairs. In this regime, the dissipative interaction is properly described by

the analytic expression given in Eq. 6.22. In contrast, for transparent tunnel junctions – in our

case for RN ≤ 150kΩ – the onset of Andreev reflections strongly enhances dissipation, which

we can phenomenologically address with a simple model based on the in-gap quasiparticle

current. However, it should be noted that another experimental study found an influence of

the circuit impedance Z (2πν) on the energy dissipation of the junction phase [142]. Analyzing

temperature-dependent measurements on the retrapping current, the authors argue that at

low temperatures quasiparticle dissipation due to thermally excited quasiparticles freezes

out, and that in this situation the dissipation is temperature-independent and dominated

by the circuit impedance. This effect could certainly also be of relevance in our experiment,

not least because our model for the retrapping current based on Cooper pair lifetime effects

overestimates this dissipation channel in comparison to independently determined lifetime

values that we obtained from fitting d I /dVJ spectra using the Maki model. However, from the

current point of view a detailed comparison of our study with Ref. [142] is difficult. Whereas

we directly tune the in-gap quasiparticle current via the tunneling resistance, the authors

of Ref. [142] tune the quasiparticle dissipation via the temperature and neglect any in-gap

dissipation due to Cooper pair lifetime effects or Andreev reflections. Nevertheless, for a better

comparability we will address the dissipative potential of our environmental impedance Z (ν)

in future work.

Last but not least, we want to address the question of why the temporal evolution of the

quantum-mechanical operator φ satisfies a classical equation of motion. Let us again assume

that the junction phase can be represented by a wavefunctionΨ(φ) and that this wavefunction

is sufficiently localized inside a potential minimum, i.e. σφ ≤ π. In the running state the

wave package moves down the potential well to a finite position where it is retrapped inside

a minimum at a particular bias current value IB . Certainly, if we repeat this measurement

over and over, the probability of presence of the junction in phase space, as expressed by

it’s wavefunction, will induce a statistical spread on the measurement result. However, a

sufficient localization ofΨ(φ) also signifies that the probability of presence is centered around

a particular point in phase space, its expectation value <φ>. In a classical picture the junction

phase therefore has a certain center of mass φc . It is now easy to imagine that the bias current

value at which this moving center of mass retraps into a potential minimum will approximately

be of the same value for a large number of measurement repetitions and, additionally, that the

statistical spread, which results from the quantum-mechanical nature of φ, will be negligible.

This attempted descriptive explanation is a close analogy to the Ehrenfest theorem that, to-

gether with the correspondence principle, describes the transition from quantum mechanics

to classical mechanics [157, 158, 144]. In 1927 Paul Ehrenfest showed for a linear potential

that the time-evolution of the expectation value of a quantum-mechanical operator obeys

98

6.8. Analysis of the Zero Voltage State

a classical equation of motion. The correspondence principle by Niels Bohr postulates that

in the limit of infinite repetitions an experiment will always converge to a result that can be

described by classical mechanics and that the statistical spread of the results will be unmea-

surable small. However, it should be emphasized at this point that the Ehrenfest theorem only

applies to systems with a linear potential and that it can actually not be applied to our system

in which the washboard potential is periodic, i.e. it is highly non-linear. Yet, it might be an

interesting question to address whether such a classical limit can also be thoretically derived

for non-linear potentials.

6.8 Analysis of the Zero Voltage State

In contrast to the phase dynamics that satisfy a classical equation of motion for our exper-

imental conditions, the analysis of the zero voltage state requires a quantum-mechanical

description, which we introduced in Sec. 6.3.2. In particular, the tunneling out of the potential

well, which is favored by a certain delocalization of the junction phase, strongly effects the

amplitude of the DC Josephson current [31, 40, 141]. For this reason the so-called switching

current IS , at which the Josephson junction leaves its zero voltage state, can serve as a valuable

probe for investigating how quantum-mechanical the junction behaves.

To this end, we determine the switching current from the current-biased experiments in the

way explained in Sec. 6.5 and plot it as a function of the tunneling resistance in Fig. 6.17(a).

For comparison, we also plot the Josepshon critical current I0, as calculated in Sec. 6.6, as a

function of RN . We find that the IS values are at least one order of magnitude smaller than

the corresponding I0 values along the entire range of RN . The double logarithmic plot of

Fig, 6.17(a) also highlights that the slope of the IS curve is twice as large as that of the I0 curve.

Since the critical current relates to the tunneling resistance as 1/RN (cf. Ch. 2, Sec. 2.2.5), the

relation between switching current and tunneling resistance should be on the order of 1/R2N .

Considering theory on the zero voltage state of a quantum-mechanical junction presented in

Sec. 6.3.2, the Zener-model indeed predicts a 1/R2N dependence for the switching current that

results from tunneling phenomena of the delocalized phase.

For a more detailed comparison of theory and experiment we employ Eq. 6.26 in order to cal-

culate the theoretical switching current, which we plot together with the experimental values

in Fig. 6.17 as a function of GN /G0. For the calculation of IS we used the fitted C J and I0 values

as determined in the voltage-biased experiments of Sec. 6.6. We find excellent agreement

between calculated and experimental values of the switching current for RN ≤ 100kΩ, which

is in accordance to reported results on planar junction geometries [141]. Only for larger values

of RN do the deviations increase. A possible explanation for this increasing deviation is illus-

trated in Fig. 6.17(b), in which we plot the relative deviation ∆IS = ∣∣(IS,exp − IS,calc)/IS,calc∣∣ as a

function of E J /EC . We find that ∆IS continuously increases when the ratio E J /EC is reduced

to values much smaller than unity. In comparison, the lower limit of the quantum-mechanical

regime that we derived in Sec. 6.2 E J /EC ≥ 0.3 is much larger than the maximum ratio for

99


Figure 6.17 – (a) Comparison of Josephson critical current I0, experimental and calculatedswitching current IS plotted as a function of RN . (b) Relative deviation ∆IS between experi-mental and calculated IS values plotted as a function of E J /EC .

100

6.8. Analysis of the Zero Voltage State

the experimental data presented in this chapter, E J /EC ≤ 0.12. Hence, for E J /EC ¿ 1 the

phase will be highly delocalized and is not a good quantum number to describe the system.

Accordingly, theoretical models, such as the Zener-model in Eq. 6.26, which are evaluated for

the regime E J ≈ EC , will fail for E J ¿ EC . In contrast, in this limit the charge is a good quantum

number and in this regime P (E)-theory should be the appropriate theory for describing our

experimental data (cf. Ch. 4) [37, 38].

In fact, in Sec. 6.6 we were able to fit the I (VJ ) curves from the corresponding voltage-biased

experiments for all values of the tunneling conductance using P (E)-theory. Moreover, for the

experiment presented in Ch. 4, Sec. 4.5 we found that P (E)-theory nicely describes our exper-

imental data for E J /EC ≤ 0.33 but fails for E J /EC > 0.7. Together these experimental results

indicate that in the quantum-mechanical regime for E J /EC > 0.3 the quantum-mechanical

nature of the junction phase significantly alters the experimental I (VJ ) curves. For smaller

values E J /EC < 0.3, however, the relevance of quantum effects is reduced, the charge evolves

as a good quantum number and P (E )-theory represents the appropriate theory for describing

our experimental data in this limit.

Figure 6.18 – Experimental and calculated values of the zero voltage resistance R0 as a functionof RN . The dashed line represents a simple square fit to the experimental data.

Before the junction switches out of the zero voltage state into the running state, eventual

phase tunneling events between adjacent potential minima are possible as well. These phase

slip events constitute a measurable effect that manifests as a finite slope in the DC Josephson

current (cf. Sec. 6.3.2) [31, 41, 139]. As a result, the Josephson effect can be observed at small

finite voltages, in our system typically on the order of ≤ 20µeV. To better understand this

phenomenon we investigate the zero voltage resistance, R0 =∆I /VJ for I < IS .

In Figure 6.18 we plot the experimental R0 values as a function of the tunneling resistance

RN . In addition, we plot the theoretical R0 values that we calculated using Eq. 6.23 [41, 139].

For the calculation we used the values for the junction capacitance C J and the Josephson

101


coupling energy E J as they were determined by analyzing the voltage-biased experiments in

Sec. 6.6. The Experimental and the calculated values of R0 match for RN ≤ 100kΩ, but exhibit

significant deviations for larger values of the tunneling resistance. Interestingly, we observe

similar deviation characteristics in the analysis of the switching current, where we find that the

Josephson junction is not in the quantum-mechanical regime, i.e. E J /EC < 0.3. Moreover, also

in literature good agreement between experimental and theoretical R0 values is only reported

for E J ≈ EC [41, 139].

The origin of these deviations in the limit E J ¿ EC comes to light when we again employ

P (E)-theory, which provides an expression to model the zero voltage resistance. In detail,

Ingold et al. could show that the zero voltage resistance of the DC Josephson effect scales

as R0 ∝ R2N [107]. In Figure 6.18 we apply a simple square fit to the experimental R0 values,

R0 = B ×R2N , and observe that R0 indeed follows such a square dependence. Moreover, in Ch. 4

we show that in the DCB regime, i.e. E J ¿ EC , thermal voltage fluctuations strongly effect the

P (E ) distribution around zero voltage. Hence, possible phase slip events due to the quantum-

mechanical nature of φ, i.e. quantum fluctuations, might be masked by thermal fluctuations.

A more detailed analysis of this zero voltage resistance was theoretically performed by Ingold

et al. in a later publication [159]. However, its application to experiment is quite involved and

will be part of future work.

6.9 Conclusion

To conclude, in this chapter we investigated the properties of a Josephson junction in a regime

in which the Josephson coupling energy is comparable to the capacitive charging energy by

means of current-biased and voltage-biased experiments. By comparing the I (VJ ) curves

of both experiments, we found that the quasiparticle tunneling currents at finite voltage

perfectly match. We also observed excellent agreement for the current around zero voltage

that originates from the coherent tunneling of Cooper pairs associated with the DC Josephson

effect. From investigations on the retrapping current IR , we found that the temporal evolution

of the junction phase satisfies a classical equation of motion. Moreover, we could determine

two different channels for energy dissipation of the junction phase. For smaller values of the

tunneling resistance, RN ≤ 150kΩ, the junction dissipates via Andreev reflections, whereas for

larger values, RN ≥ 150kΩ, the energy dissipation is dominated by lifetime effects of Cooper

pairs.

In contrast to the above picture, when investigating the switching current IS we found that an

appropriate description of the zero voltage state stability requires a fully quantum-mechanical

picture. For E J < EC find that the premature switching out of the zero voltage state can

be explained by phase tunneling through the potential walls, although the junction is not

directly operated in the quantum-mechanical regime, as defined in Sec. 2.5. In contrast, in the

limit E J ¿ EC we find that the relevance of phase tunneling phenomena is reduced and that

P (E)-theory represents the appropriate theory to describe our experimental data.

102

7 Conclusion & Outlook

7.1 Conclusion

The aim of this thesis was to characterize the properties of a Josephson junction in an STM at

millikelvin temperatures and to implement JSTM as a versatile probe on the atomic scale. Such

a process first demands an understanding of the regime in which the junction is operated.

Second, on this basis the inelastic interaction of the Josephson junction with its immedi-

ate environment as well as the dynamics of the Josephson junction have to be understood.

Moreover, an appropriate theory has to be implemented that allows us to extract the critical

Josephson current from experimental data in view of the application of JSTM as a probe for

superconductivity.

To this end we investigated the I (VJ) tunneling characteristics of the Josephson junction in our

STM at a base temperature of 15 mK by means of current-biased and voltage-biased experi-

ments. In Ch. 4 we observed that in the tunnel regime, GN ¿G0, the Josephson junction is

operated in the dynamical Coulomb blockade (DCB) regime in which the sequential tunneling

of Cooper pairs dominates the tunneling current. Employing P (E)-theory [37, 38, 112, 107]

allowed us to model experimental I (VJ) characteristics from voltage-biased experiments and

determine experimental values of the Josephson critical current in agreement to theory [8].

Moreover, we observed a breakdown of P (E)-theory for experiments at large tunneling con-

ductance GN ≈G0, which could indicate that the coherent tunneling of Cooper pairs strongly

contributes to the tunneling current in this limit.

Analyzing the I (VJ) curves from voltage-biased experiments we observed in Ch. 5 that the

Josephson junction in an STM at millikelvin temperatures is highly sensitive to its electro-

magnetic environment [39]. The combination of the experiment with finite integral sim-

ulations revealed that the immediate environment of a Josephson junction in an STM is

frequency-dependent and additionally, that the STM geometry shares the electric properties

of a monopole antenna with the STM tip acting as the antenna.

Comparing the I (VJ) curves of voltage-biased and current-biased experiments in Ch. 6 we ob-

103

Chapter 7. Conclusion & Outlook

served that the dynamics of the junction phase exhibit underdamped characteristics. We could

show that at these low temperatures the dynamics of the junction phase are dominated by the

in-gap quasiparticle tunneling current. For opaque junctions GN < 0.1G0 this quasiparticle

current originates from Cooper pair lifetime effects and that in more transparent junctions,

GN > 0.1G0, it is strongly enhanced by Andreev reflections. Moreover, from comparing current-

and voltage-biased experiments we also observed that the quantum-mechanical nature of the

junction phase strongly effects the junction properties for GN ≤G0.

To conclude, we fully characterized the properties of the Josephson junction in an STM that is

operated at millikelvin temperatures. Hence, this work represents necessary and fundamental

steps that allow us to employ the Josephson effect as a versatile probe on the atomic scale. In

the following outlook, we will give examples for JSTM applications that can be realized using

our experiment. Moreover, we will give an outlook on general questions of Josephson tunneling

which can be ideally addressed using our STM experiment. Partly, the given examples directly

follow from the thesis’ results.

7.2 Outlook

7.2.1 JSTM: Probing the Superconducting Ground State

A very promising application for Josephson STM is to study spatial variations of the supercon-

ducting order parameter ∆ in non-conventional superconductors. In BSCCO, for example,

the superconducting gap, as measured via quasiparticle excitations, reveals a strong response

to local doping by both magnetic and non-magnetic impurities [74, 160, 101, 161]. Here, the

sensitivity of JSTM to the superconducting ground state itself could provide more insight

into the underlying interaction. Moreover, for the same class of superconductors JSTM could

help to shed light onto the pairing symmetry and pairing mechanism of the superconducting

ground state as proposed by the group of Alex Balatsky [14].

Another interesting question to explore is the perturbation of the superconducting ground

state of a BCS superconductor by a single magnetic impurity. While theory predicts a strong

response of the superconducting ground state [18], in experiments no change of quasiparticle

excitation energy was observed [16, 19]. Even in experiments where the interaction between

the magnetic impurity and the substrate was evidently strong, the gap remained unaffected

[20]. Again, JSTM could represent the appropriate experimental technique to study the local

interaction of a superconductor with a magnetic impurity, as it was proposed by theory [17]. If

the interaction between magnetic impurity and superconductor is sufficiently strong, even a

π-junction where the order parameter of the sample changes sign, ∆→−∆, could be realized

[17].

We recently started with first experiments on the interaction of a single magnetic moment

with a superconducting ground state. To this end, we deposited copper phtalocyanine (CuPc)

104

7.2. Outlook

molecules – a spin-1/2 system – on top of the vanadium surface 1. Fig. 7.1(a) displays a single

CuPc molecule on top of a V(001) surface. We already performed first test measurements

on this sample system and we could observe the Josephson effect through a single molecule

as shown in Fig. 7.1(b). Moreover, we were also able to fit these initial I (VJ) curves by using

P (E)-theory.

Figure 7.1 – (a) Single CuPc molecule on top of a V(001) surface. (b) Large scale: Initialmeasurement of the Josephson effect through a single CuPc molecule. The inset shows aP (E)-fit to the experimental I (VJ) curve.

These first results indicate that it is indeed possible to measure and analyze the Cooper pair

current through a single adsorbate on a superconducting surface, an observation that provides

the basis for a more detailed study.

7.2.2 Cooper Pairs and Photons I: The AC Josephson Spectrometer

Large research efforts concentrate on studying single electronic and nuclear spins using

a variety of experimental techniques such as STM, break junctions or NV magnetometry

[24, 129, 162, 163, 164]. The sensitivity of the Josephson junction in our STM to its electro-

magnetic environment could also be exploited in the AC Josephson spectrometer for probing

electromagnetic transitions in nanoscale systems, a concept that has already been realized

in planar junction geometries [22, 23]. A possible sample system that can be prepared in our

experiment are single magnetic molecules, such as CuPc molecules shown in the last section.

However, in the case of CuPc on V(001) we could not observe inelastic spin-flip transitions in

magnetic fields up to 7T, which indicates that the spin moment of the copper atom could have

been compensated by the substrate due to hybridization effects. However, it was shown by

the group of Katharina Franke that ligands other than phtalocyanine can decouple the metal

1The CuPc molecules were deposited by means of thermal evaporation at a temperature of T = 385C underUHV conditions. During evaporation the sample was thermalized to T = 20C.

105


center from a superconducting substrate such, that the magnetic moment is preserved and

magnetic transitions are observable [164].

In addition, the relatively high spectral resolution of the AC Josephson spectrometer of about

12µeV in our current setup should allow us to study hyperfine transitions of a single nu-

clear spin. The hyperfine transition of an individual Bismuth atom, for instance, has been

calculated to be ∆E ≈ 25µeV [25]. This transition clearly matches our spectral resolution.

Moreover, the high sensitivity of the AC Josephson spectrometer [22], in conjunction with the

superconducting gap, favors detection of tiny signals as expected for this experiment [25].

7.2.3 Cooper Pairs and Photons II: Nonlinear Cooper Pair Tunnel Effects

In recent years the combination of a Josephson junction and a resonator in its immediate envi-

ronment served as a valuable tool for theoretical and experimental studies of the Cooper-pair

photon interaction [105], circuit quantum electrodynamics [126] and non-classical photon

generation [127]. Of particular interest for us is a theoretical work by the group of Joachim

Ankerhold, in which they investigate the non-linear interaction between a photon-field in

a resonator and a Josephson junction [106]. By tuning both the resonator properties, i.e.

the resonance frequency ν0 and the quality factor Q, and the junction properties, i.e. the

Josephson coupling energy EJ and the capacitive charging energy EC, it should be possible

to observe the transition from linear to nonlinear interactions between a resonator and a

Josephson junction. Linear interactions correspond to a regime in which inelastic Cooper

pair tunneling events excite the resonator modes but where the tunneling rate is so small,

that for subsequent tunneling events the resonator will be always in the ground state again.

However, if the Cooper pair tunneling rate or the quality of the resonator are increased, the

resonator is excited faster than it can relax back into the ground state. In this situation, the

occupied resonator can interact with the Josephson junction, which results in non-linear

coupling effects such as charge-photon correlations [106].

Our setup also hosts the combination of a Josephson junction with a resonator, which is

formed by the λ/4-resonator in the tip. Moreover, we can tune both the resonator frequency

ν0 and its quality factor Q by changing the tip holder geometry as well as the Josephson

coupling energy for more than two orders of magnitude by changing the tip-sample distance.

To investigate whether our system allows us to observe this transition between linear and

nonlinear coupling, we investigate the coupling parameter β= ln(2EJECQ/(ħν0)2), as given

in Ref. [106]. For a particular ratio of EC versus ν0 this parameter defines whether linear or

nonlinear coupling occurs. In Fig. 7.2 we plot calculated values of β as a function of the

normalized tunneling conductance GN/G0. For the calculation we used a parameter set that

is typical for our experiment and a ratio ln(EC/(ħν0)) ≈ −0.2 (∆1 = 800µeV, ∆2 = 400µeV,

ħν0 = 156µeV that corresponds to an l = 2mm long STM tip, Q = 4 and CJ = 2.5fF) [106]. We

also plot the transition point from linear to nonlinear junction resonator interactions that we

calculated for the ratio ln(EC/(ħν0)) ≈−0.2 according to Ref. [106]. We find that the transition

106

7.2. Outlook

Figure 7.2 – Coupling parameter β as a function of the normalized tunneling conductanceGN/G0.

from linear to nonlinear interactions between Josephson junction and resonator occurs well

within the tunnel regime GN ¿G0. For this reason, we should be able to observe a change

in the experimental I (VJ) characteristics if we cross this transition point by tuning EJ via the

tip-sample distance. In detail, for the nonlinear interaction regime we expect deviations

from the E 2J -dependence of the resonant current peak, which originates from the inelastic

interaction with the resonator at 2eVJ = hν0, as it is observed for linear interactions in the DCB

regime [37, 38, 107].

It should be noted that, before the theoretical work on this problem [106] was published, we

already performed experiments in which the shown parameter ratio, ln(EC/(ħν0)) ≈−0.2, was

given (cf. Ch. 5). Hence, it should be straightforward for us to repeat this experiment and

address this exciting problem in more detail.

7.2.4 Tuning the Energy Scales: Coherent vs. Sequential Cooper pair Tunneling

For a Josephson junction in which the Josephson coupling energy is much smaller than the

capacitive charging energy of the junction, EJ ¿ EC, the tunneling current is carried by the

sequential tunneling of Cooper pairs and can be described by P (E )-theory [37, 38], whereas in

the opposite limit, where EJ À EC, coherent tunneling of Cooper pairs sustain the tunneling

current [7, 9]. By analyzing experimental I (VJ ) curves, we indeed find that in the limit EJ ≈ EC

P (E )-theory fails to describe our experimental data around zero voltage, indicating an increas-

ing contribution of coherent Cooper pair tunneling to the tunneling current. This transition

between sequential and coherent tunneling had already been addressed experimentally at

the end of the 1980s [141, 165]. However, this transition had never been clearly resolved, most

likely due to the experimental constraint that in experiments using planar junction geometries

it is difficult to change the ratio EJ versus EC. Until now, we were also not able to observe the

107


full transition to the coherent tunneling regime because for E J ≈ EC we already reached point

contact GN/G0 ≥ 1 (cf. Ch. 4).

Figure 7.3 – Comparison of EJ and EC as a function of the normalized tunneling conductanceGN/G0.

In order to address this transition in another experiment we therefore have to either increase

EJ by changing the electrode material to superconductors of larger order parameter ∆, or

decrease EC by a larger junction capacitance CJ. For our STM setup the latter approach

appears more favorable, since a larger junction capacitance can be obtained by varying the tip

diameter and shape. Figure 7.3 compares calculated values of EJ and EC as a function of the

normalized tunneling conductance GN/G0 for typical experimental parameter (∆1 = 800µeV,

∆2 = 400µeV) and assuming a junction capacitance of 50 fF. We find that the transition point

EJ = EC has moved deeply into the tunnel regime. Simple calculations show that such an

increased junction capacitance is readily obtained by tuning the thickness of the tip and

shaping the sub-millimeter geometry of the tip apex. For this reason, we should be able to

observe the full transition from sequential to coherent Cooper pair tunneling in our JSTM

setup.

7.2.5 Studying the Quasiparticle Dissipation of Josephson Junctions

The decoherence of prepared qubit states via dissipative interaction with the environment is a

fundamental problem in quantum computing. In Josephson phase qubits, the decoherence of

the junction phase can result, for instance, from dissipative quasiparticle excitations, for which

reason this phenomenon has attracted significant research interest [166, 167, 168, 169, 170].

As we show in Ch. 6 of this thesis, JSTM at millikelvin temperatures enables detailed studies

of dissipation processes in Josephson junctions. First, at these low temperatures thermal

quasiparticle excitations, a possible yet difficult-to-control source of dissipation, are frozen

out and secondly, changing the tip sample distance allows us to precisely tune the amplitude of

108

7.2. Outlook

dissipative quasiparticle excitations. For a Josephson junction that consists of two vanadium

electrodes we find that the minimum dissipation is given by lifetime effects of Cooper pairs,

which result from impurity scattering effects or surface scattering effects in the confined tip

geometry [156], for instance. In order to minimize these lifetime effects, it is desirable to

employ a different junction material. For this reason we recently started working on junctions

that consist of an aluminum (001) crystal as the sample and an amorphous aluminum wire as

the STM tip. In former experiments we already found that lifetime effects in such aluminum

STM tips are negligibly small [81, 35].

Such a Josephson junction should allow us to investigate quasiparticle dissipation phenomena

in more detail and we might approach the minimum dissipation limit as given by Chen,

Fisher and Leggett in Ref. [147]. Moreover, it should help us to study the in-gap quasiparticle

dissipation due to Andreev reflections in more detail, which we could already observe in

vanadium junctions (cf. Ch. 6). Hence, JSTM helps us to shed more light on the quasiparticle

dissipation phenomena in Josephson junctions and in this way, we can potentially contribute

to the highly discussed topic of decoherence phenomena in Josephson phase qubits.

109

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Josephson-junction qubits from nonequilibrium quasiparticle excitations. Phys. Rev.

Lett., 103:097002, Aug 2009.

[170] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret.

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ticles. Nature, 508(7496):369–372, 2014.

122

Curriculum Vitae

Personal Details

Name Berthold Jäck

Date of Birth February 21, 1985

Place of Birth Karlsruhe, Germany

Nationality German

Education

Since May 2012 PhD student

Ecole Polytechnique Fédérale de Lausanne

Doctoral School of Physics

Thesis: Josephson Tunneling at the Atomic Scale:

The Josephson Effect as a Local Probe

Aug. 2008 - May 2009 Student Exchange Year

University of California at Berkeley

Fellow of the German Academic Exchange Association (DAAD)

Oct. 2005 - May 2011 Diploma Studies of Nanotechnology

University of Würzburg, Würzburg

Diploma degree with distinction

Thesis: Exciton Dynamics in Ordered and Disordered

Organic Systems

Sep. 1995 - Jul. 2004 High School

Gymnasium Karlsbad, Karlsbad

123

Bibliography

Scientific & Working Experience

Since Oct. 2011 Research Associate

Max-Planck-Institute for Solid State Research, Stuttgart

Oct. 2009 - Feb. 2011 Teaching Assistant

University of Würzburg, Würzburg

Tutoring of lectures and lab classes

Aug. 2008 - May 2009 Research Internship

Lawrence Berkeley National Laboratory, USA

Preparation & characterization of Graphene samples

Feb. 2008 - Apr- 2008 Research Internship

Qimonda Dresden GmbH, Dresden

Technology development in semiconductor industry

May 2005 - Sep. 2005 Advisor

School for Disabled Children, Karlsbad

Nursing of disabled children

Sep. 2004 - May 2005 Civil Service

School for Disabled Children, Karlsbad

Nursing of disabled children

124

Publications

• B. Jäck, J. Senkpiel, M. Eltschka, M. Etzkorn, C. R. Ast, and K. Kern, “Quasiparticle

dissipation phenomena in Josepson junctions“, in preparation (2015)

• M. Eltschka, B. Jäck, M. Assig, O. V. Kondrashov, M. A. Skvortsov, M. Etzkorn, C. R. Ast,

and K. Kern, “Superconducting STM tips in a magnetic field: geometry-controlled order

of the phase transition“, submitted (2015)

• B. Jäck, M. Eltschka, M. Assig, M. Etzkorn, C. R. Ast, and K. Kern, “Critical Josephson

current in the dynamical Coulomb blockade regime“, submitted (2015)

• B. Jäck, M. Eltschka, M. Assig, A. Hardock, M. Etzkorn, C. R. Ast, and K. Kern, “A

nanoscale gigahertz source realized with Josephson scanning tunneling microscopy“,

Appl. Phys. Lett. 106, 013109 (2015)

• M. Eltschka, B. Jäck, M. Assig, O. V. Kondrashov, M. A. Skvortsov, M. Etzkorn, C. R. Ast,

and K. Kern, “Probing Absolute Spin Polarization at the Nanoscale“, Nano Lett. 14, 7171

(2014)

• B. Gieseking, B. Jäck, E. Preis, S. Jung, M. Forster, U. Scherf, C. Deibel, and V. Dyakonov,

“Excitation Dynamics in Low Band Gap Donor–Acceptor Copolymers and Blends“, Adv.

Energy. Mater. 2 (12), 1477 – 1482 (2012)

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